ROBUST CONTROL, THEORY AND APPLICATIONS Edited by Andrzej Bartoszewicz INTECHWEB.ORG Robust Control, Theory and Applications Edited by Andrzej Bartoszewicz Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. 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ISBN 978-953-307-229-6 OPEN ACCESS PUBLISHER INTECH INTECHopen free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface XI Part 1 Fundamental Issues in Robust Control 1 Chapter 1 Introduction to Robust Control Techniques 3 Khaled Halbaoui, Djamel Boukhetala and Fares Boudjema Chapter 2 Robust Control of Hybrid Systems 25 Khaled Halbaoui, Djamel Boukhetala and Fares Boudjema Chapter 3 Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 43 Rama K. Yedavalli and Nagini Devarakonda Part 2 H-infinity Control 67 Chapter 4 Robust H, PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 69 Endra Joelianto Chapter 5 Robust H w Tracking Control of Stochastic Innate Immune System Under Noises 89 Bor-Sen Chen, Chia-Hung Chang and Yung-Jen Chuang Chapter 6 Robust H w Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 117 Ronghao Wang, Jianchun Xing, Ping Wang, Qiliang Yang and Zhengrong Xiang Part 3 Sliding Mode Control 139 Chapter 7 Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 141 Hai-Ping Pang and Qing Yang VI Contents Chapter 8 Robust Delay-Independent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 163 Elbrous M. Jafarov Chapter 9 A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 197 Masanao Obayashi, Norihiro Nakahara, Katsumi Yamada, Takashi Kuremoto, Kunikazu Kobayashi and Liangbing Feng Part 4 Selected Trends in Robust Control Theory 215 Chapter 10 Robust Controller Design: New Approaches in the Time and the Frequency Domains 217 Vojtech Vesely, Danica Rosinova and Alena Kozakova Chapter 11 Robust Stabilization and Discretized PID Control 243 Yoshifumi Okuyama Chapter 12 Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 261 Makoto Katoh Chapter 13 Passive Fault Tolerant Control 283 M. Benosman Chapter 14 Design Principles of Active Robust Fault Tolerant Control Systems 309 Anna Filasova and Dusan Krokavec Chapter 15 Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 339 Alejandro H. Gonzalez and Darci Odloak Chapter 16 Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 371 Shuxiang Guo Chapter 17 A Frequency Domain Quantitative Technique for Robust Control System Design 391 Jose Luis Guzman, Jose Carlos Moreno, Manuel Berenguel, Francisco Rodriguez and Julian Sanchez-Hermosilla Chapter 18 Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 423 Sabato Manfredi Contents VII Chapter 19 On Stabilizability and Detectability of Variational Control Systems 441 Bogdan Sasu and Adina Luminita Sasu Chapter 20 Robust Linear Control of Nonlinear Flat Systems 455 Hebertt Sira- Ramirez, John Cortes- Romero and Alberto Luviano-Juarez Part 5 Robust Control Applications 477 Chapter 21 Passive Robust Control for Internet-Based Time-Delay Switching Systems 479 Hao Zhang and Huaicheng Yan Chapter 22 Robust Control of the Two-mass Drive System Using Model Predictive Control 489 Krzysztof Szabat, Teresa Ortowska-Kowalska and Piotr Serkies Chapter 23 Robust Current Controller Considering Position Estimation Error for Position Sensor-less Control of Interior Permanent Magnet Synchronous Motors under High-speed Drives 507 Masaru Hasegawa and Keiju Matsui Chapter 24 Robust Algorithms Applied for Shunt Power Quality Conditioning Devices 523 Joao Marcos Kanieski, Hilton Abilio Grundling and Rafael Cardoso Chapter 25 Robust Bilateral Control for Teleoperation System with Communication Time Delay - Application to DSD Robotic Forceps for Minimally Invasive Surgery - 543 Chiharu Ishii Chapter 26 Robust Vehicle Stability Control Based on Sideslip Angle Estimation 561 Haiping Du and Nong Zhang Chapter 27 QFT Robust Control of Wastewater Treatment Processes 577 Marian Barbu and Sergiu Caraman Chapter 28 Control of a Simple Constrained MIMO System with Steady-state Optimization 603 Frantisek Dusek and Daniel Hone Chapter 29 Robust Inverse Filter Design Based on Energy Density Control 619 Junho Lee and Young-Cheol Park VIII Contents Chapter 30 Robust Control Approach for Combating the Bullwhip Effect in Periodic-Review Inventory Systems with Variable Lead-Time 635 Przemystaw Ignaciuk and Andrzej Bartoszewicz Chapter 31 Robust Control Approaches for Synchronization of Biochemical Oscillators 655 Hector Puebla, Rogelio Hernandez Suarez, Eliseo Hernandez Martinez and Margarita M. Gonzalez-Brambila Preface The main purpose of control engineering is to steer the regulated plant in such a way that it operates in a required manner. The desirable performance of the plant should be obtained despite the unpredictable influence of the environment on all parts of the control system, including the plant itself, and no matter if the system designer knows precisely all the parameters of the plant. Even though the parameters may change with time, load and external circumstances, still the system should preserve its nominal properties and ensure the required behaviour of the plant. In other words, the princi- pal objective of control engineering is to design control (or regulation) systems which are robust with respect to external disturbances and modelling uncertainty. This ob- jective may very well be obtained in a number of ways which are discussed in this monograph. The monograph is divided into five sections. In section 1 some principal issues of the field are presented. That section begins with a general introduction presenting well developed robust control techniques, then discusses the problem of robust hybrid con- trol and concludes with some new insights into stability and control of linear interval parameter plants. These insights are made both from an engineering (quantitative) perspective and from the population (community) ecology point of view. The next two sections, i.e. section 2 and section 3 are devoted to new results in the framework of two important robust control techniques, namely: H-infmity and sliding mode control. The two control concepts are quite different from each other, however both are nowadays very well grounded theoretically, verified experimentally, and both are regarded as fundamental design techniques in modern control theory. Section 4 presents various other significant developments in the theory of robust control. It begins with three contributions related to the design of continuous and discrete time robust proportional integral derivative controllers. Next, the section discusses selected problems in pas- sive and active fault tolerant control, and presents some important issues of robust model predictive and fuzzy control. Recent developments in quantitative feedback theory, stabilizability and detectability of variational control systems, control of multi agent systems and control of flat systems are also the topics considered in the same section. The monograph is concerned not only with a wide spectrum of theoretical issues in robust control domain, but it also demonstrates a number of successful, re- cent engineering and non-engineering applications of the theory. These are described in section 5 and include internet based switching control, and applications of robust XII Preface control techniques in electric drives, power electronics, bilateral teleoperation systems, automotive industry, wastewater treatment, thermostatic baths, multi-channel sound reproduction systems, inventory management and biological processes. In conclusion, the main objective of this monograph is to present a broad range of well worked out, recent theoretical and application studies in the field of robust control system analysis and design. We believe, that thanks to the authors and to the Intech Open Access Publisher, this ambitious objective has been successfully accomplished. The editor and authors truly hope that the result of this joint effort will be of signifi- cant interest to the control community and that the contributions presented here will advance the progress in the field, and motivate and encourage new ideas and solutions in the robust control area. Andrzej Bartoszewicz Institute of Automatic Control, Technical University of Lodz Poland Parti Fundamental Issues in Robust Control 1 Introduction to Robust Control Techniques Khaled Halbaoui 1 ' 2 , Djamel Boukhetala 2 and Fares Boudjema 2 2 Power Electronics Laboratory, Nuclear Research Centre of Brine CRNB, BP 180 Ainoussera 1 7200, Djelfa 2 Laboratoire de Commande des Processus, ENSP, 10 avenue Pasteur, Hassan Badi, BP 182 El-Harrach Algeria 1. Introduction The theory of "Robust" Linear Control Systems has grown remarkably over the past ten years. Its popularity is now spreading over the industrial environment where it is an invaluable tool for analysis and design of servo systems. This rapid penetration is due to two major advantages: its applied nature and its relevance to practical problems of automation engineer. To appreciate the originality and interest of robust control tools, let us recall that a control has two essential functions: • shaping the response of the servo system to give it the desired behaviour, • maintaining this behaviour from the fluctuations that affect the system during operation (wind gusts for aircraft, wear for a mechanical system, configuration change to a robot.). This second requirement is termed "robustness to uncertainty". It is critical to the reliability of the servo system. Indeed, control is typically designed from an idealized and simplified model of the real system. To function properly, it must be robust to the imperfections of the model, i.e. the discrepancies between the model and the real system, the excesses of physical parameters and the external disturbances. The main advantage of robust control techniques is to generate control laws that satisfy the two requirements mentioned above. More specifically, given a specification of desired behaviour and frequency estimates of the magnitude of uncertainty, the theory evaluates the feasibility, produces a suitable control law, and provides a guaranty on the range of validity of this control law (strength). This combined approach is systematic and very general. In particular, it is directly applicable to Multiple-Input Multiple Output systems. To some extent, the theory of Robust Automatic Control reconciles dominant frequency (Bode, Nyquist, PID) and the Automatic Modern dominated state variables (Linear Quadratic Control, Kalman). It indeed combines the best of both. From Automatic Classic, it borrows the richness of the frequency analysis systems. This framework is particularly conducive to the specification of performance objectives (quality of monitoring or regulation), of band-width and of robustness. From Automatic Modern, it inherits the simplicity and power of synthesis Robust Control, Theory and Applications methods by the state variables of enslavement. Through these systematic synthesis tools, the engineer can now impose complex frequency specifications and direct access to a diagnostic feasibility and appropriate control law. He can concentrate on finding the best compromise and analyze the limitations of his system. This chapter is an introduction to the techniques of Robust Control. Since this area is still evolving, we will mainly seek to provide a state of the art with emphasis on methods already proven and the underlying philosophy. For simplicity, we restrict to linear time invariant systems (linear time-invariant, LTI) continuous time. Finally, to remain true to the practice of this theory, we will focus on implementation rather than on mathematical and historical aspects of the theory. 2. Basic concepts The control theory is concerned with influencing systems to realize that certain output quantities take a desired course. These can be technical systems, like heating a room with output temperature, a boat with the output quantities heading and speed, or a power plant with the output electrical power. These systems may well be social, chemical or biological, as, for example, the system of national economy with the output rate of inflation. The nature of the system does not matter. Only the dynamic behaviour is of great importance to the control engineer. We can describe this behaviour by differential equations, difference equations or other functional equations. In classical control theory, which focuses on technical systems, the system that will be influenced is called the [controlled) plant. In which kinds in manners can we influence the system? Each system is composed not only of output quantities, but as well of input quantities. For the heating of a room, this, for example, will be the position of the valve, for the boat the power of the engine and angle of the rudder. These input variables have to be adjusted in a manner that the output variables take the desired course, and they are called actuating variables. In addition to the actuating variables, the disturbance variables affect the system, too. For instance, a heating system, where the temperature will be influenced by the number of people in the room or an open window, or a boat, whose course will be affected by water currents. The desired course of output variables is defined by the reference variables. They can be defined by operator, but they can also be defined by another system. For example, the autopilot of an aircraft calculates the reference values for altitude, the course, and the speed of the plane. But we do not discuss the generation of reference variables here. In the following, we take for them for granted. Just take into account that the reference variables do not necessarily have to be constant; they can also be time-varying. Of which information do have we need to calculate the actuating variables to make the output variables of the system follow the variables of reference? Clearly the reference values for the output quantities, the behavior of the plant and the time-dependent behavior of the disturbance variables must be known. With this information, one can theoretically calculate the values of the actuating variables, which will then affect the system in a way that the output quantities will follow the desired course. This is the principle of a steering mechanism (Fig. 1). The input variable of the steering mechanism is the reference variable co , its output quantity actuating variable u , which again - with disturbance variable w forms the input value of the plant, y represents the output value of the system. The disadvantage of this method is obvious. If the behavior of the plant is not in accordance with the assumptions which we made about it, or if unforeseen disruptions, then the Introduction to Robust Control Techniques quantities of output will not continue to follow the desired course. A steering mechanism cannot react to this deviation, because it does not know the output quantity of the plant. Steering ;j-^C>-> ^ Plant ^ ~K$ Fig. 1. Principle of a steering mechanism A improvement which can immediately be made is the principle of an (automatic) control (Fig. 2). Inside the automatic check, the reference variable co is compared with the measured output variable of the plant y (control variable), and a suitable output quantity of the controller u (actuating variable) are calculated inside the control unit of the difference Ay (control error). During old time the control unit itself was called the controller, but the modern controllers, including, between others, the adaptive controllers (Boukhetala et al., 2006), show a structure where the calculation of the difference between the actual and wished output value and the calculations of the control algorithm cannot be distinguished in the way just described. For this reason, the tendency today is towards giving the name controller to the section in which the variable of release is obtained starting from the reference variable and the measured control variable. \J— ► Controller ►CJ) — ► Actuator p> Process — ►Of-^ Metering Element j Fig. 2. Elements of a control loop The quantity u is usually given as low-power signal, for example as a digital signal. But with low power, it is not possible to tack against a physical process. How, for example, could be a boat to change its course by a rudder angle calculated numerically, which means a sequence of zeroes and ones at a voltage of 5 V? Because it's not possible directly, a static inverter and an electric rudder drive are necessary, which may affect the rudder angle and the boat's route. If the position of the rudder is seen as actuating variable of the system, the static inverter, the electric rudder drive and the rudder itself from the actuator of the system. The actuator converts the controller output, a signal of low power, into the actuating variable, a signal of high power that can directly affect the plant. Alternatively, the output of the static inverter, that means the armature voltage of the rudder drive, could be seen as actuating variable. In this case, the actuator would consist only of static converter, whereas the rudder drive and the rudder should be added to the plant. These various views already show that a strict separation between the actuator and the process is not possible. But it is not necessary either, as for the design of the controller; Robust Control, Theory and Applications we will have to take every transfer characteristic from the controller output to the control variable into account anyway. Thus, we will treat the actuator as an element of the plant, and henceforth we will employ the actuating variable to refer to the output quantity of the controller. For the feedback of the control variable to the controller the same problem is held, this time only in the opposite direction: a signal of high power must be transformed into a signal of low power. This happens in the measuring element, which again shows dynamic properties that should not be overlooked. Caused by this feedback, a crucial problem emerges, that we will illustrate by the following example represented in (Fig. 3). We could formulate strategy of a boat's automatic control like this: the larger the deviation from the course is, the more the rudder should be steered in the opposite direction. At a glance, this strategy seems to be reasonable. If for some reason a deviation occurs, the rudder is adjusted. By steering into the opposite direction, the boat receives a rotatory acceleration in the direction of the desired course. The deviation is reduced until it disappears finally, but the rotating speed does not disappear with the deviation, it could only be reduced to zero by steering in the other direction. In this example, because of the rotating speed of the boat will receive a deviation in the other direction after getting back to the desired course. This is what happened after the rotating speed will be reduced by counter-steering caused by the new deviation. But as we already have a new deviation, the whole procedure starts again, only the other way round. The new deviation could be even greater than the first. The boat will begin zigzagging its way, if worst comes to worst, with always increasing deviations. This last case is called instability. If the amplitude of vibration remains the same, it is called borderline of stability. Only if the amplitudes decrease the system is stable. To receive an acceptable control algorithm for the example given, we should have taken the dynamics of the plant into account when designing the control strategy. A suitable controller would produce a counter-steering with the rudder right in time to reduce the rotating speed to zero at the same time the boat gets back on course. Desired Qourse_y^^_^^^^^_ Fig. 3. Automatic cruise control of a boat This example illustrates the requirements with respect to the controlling devices. A requirement is accuracy, i.e. the control error should be also small as possible once all the initial transients are finished and a stationary state is reached. Another requirement is the speed, i.e. in the case of a changing reference value or a disturbance; the control error should be eliminated as soon as possible. This is called the response behavior. The requirement of the third and most important is the stability of the whole system. We will see that these conditions are contradicted, of this fact of forcing each kind of controller (and therefore fuzzy controllers, too) to be a compromise between the three. Introduction to Robust Control Techniques 3. Frequency response If we know a plant's transfer function, it is easy to construct a suitable controller using this information. If we cannot develop the transfer function by theoretical considerations, we could as well employ statistical methods on the basis of a sufficient quantity of values measured to determine it. This method requires the use of a computer, a plea which was not available during old time. Consequently, in these days a different method frequently employed in order to describe a plant's dynamic behavior, frequency response (Franklin et al., 2002). As we shall see later, the frequency response can easily be measured. Its good graphical representation leads to a clear method in the design process for simple PID controllers. Not to mention only several criteria for the stability, which as well are employed in connection with fuzzy controllers, root in frequency response based characterization of a plant's behavior. The easiest way would be to define the frequency response to be the transfer function of a linear transfer element with purely imaginary values for s. Consequently, we only have to replace the complex variable s of the transfer function by a variable purely imaginary, jco : G(jco) = G(s)| . . The frequency response is thus a complex function of the parameter co . Due to the restriction of s to purely imaginary values; the frequency response is only part of the transfer function, but a part with the special properties, as the following theorem shows: Theorem 1 If a linear transfer element has the frequency response G(jco) , then its response to the input signal x(t) = asincot will be-after all initial transients have settled down-the output signal y(t) = a\G(ja>)\sin(cot + <p(G(ja>))) (1) If the following equation holds: ]\ g (t)\dt- (2) |G(;&>)| is obviously the ratio of the output sine amplitude to the input sine amplitude ((transmission) gain or amplification). 0(G(jco) is the phase of the complex quantity G(jco) and shows the delay of the output sine in relation to the input sine (phase lag). g(t) is the impulse response of the plant. In case the integral given in (2) does not converge, we have to add the term r(t) to the right hand side of (I), which will, even for t -^ oo , not vanish. The examination of this theorem shows clearly what kind of information about the plant the frequency response gives: Frequency response characterizes the system's behavior for any frequency of the input signal. Due to the linearity of the transfer element, the effects caused by single frequencies of the input signal do not interfere with each other. In this way, we are now able to predict the resulting effects at the system output for each single signal component separately, and we can finally superimpose these effects to predict the overall system output. Unlike the coefficients of a transfer function, we can measure the amplitude and phase shift of the frequency response directly: The plant is excited by a sinusoidal input signal of a certain frequency and amplitude. After all initial transients are installed we obtain a sinusoidal signal at the output plant, whose phase position and amplitude differ from the input signal. The quantities can be measured, and depending to (1), this will also instantly 8 Robust Control, Theory and Applications provide the amplitude and phase lag of the frequency response G(jco) . In this way, we can construct a table for different input frequencies that give the principle curve of the frequency response. Take of measurements for negative values of co , i.e. for negative frequencies, which is obviously not possible, but it is not necessary either, delay elements for the transfer functions rational with real coefficients and for G(jco) will be conjugate complex to G(-jco) . Now, knowing that the function G(jco) for co > already contains all the information needed, we can omit an examination of negative values of co . 4. Tools for analysis of controls 4.1 Nyquist plot A Nyquist plot is used in automatic control and signal processing for assessing the stability of a system with feedback. It is represented by a graph in polar coordinates in which the gain and phase of a frequency response are plotted. The plot of these phasor quantities shows the phase as the angle and the magnitude as the distance from the origin (see. Fig.4). The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 5 I ( I -0.! OdB -2dB ~~~^\ ^ '/S \ 2dB '■' / -4 dB ^ 4dB 6dB -6dB N "10 dB -1,6 dB \ \ r n — il+ q> l ! _ J_ _ - - , 7 / I \ v / I \ 7 %- J / / " -- \ z x ^ ^ <^~- - "" ^^ Nyquist Diagram 40 - /^ OdB ^"^\. x - 30 " ( V 20 ' \ /^H^^^ j w 10 1 1 E -10 " V - \V| dB '-2_dB 2 ^ / . -20 " ( ^^^ \ - -30 - V -40 ^^^ ^^^^ / First-order system Second-order systems Fig. 4. Nyquist plots of linear transfer elements Assessment of the stability of a closed-loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. the same system without its feedback loop). This method is easily applicable even for systems with delays which may appear difficult to analyze by means of other methods. Nyquist Criterion: We consider a system whose open loop transfer function (OLTF) is G(s) ; when placed in a closed loop with feedback H(s) , the closed loop transfer function (CLTF) G then becomes The case where H = 1 is usually taken, when investigating stability, 1 + G.H and then the characteristic equation, used to predict stability, becomes G + 1 = . We first construct The Nyquist Contour, a contour that encompasses the right-half of the complex plane: • a path traveling up the jco axis, from -700 to + 700 . • a semicircular arc, with radius r -^ 00 , that starts at + jco and travels clock-wise to -700 Introduction to Robust Control Techniques The Nyquist Contour mapped through the function 1 + G(s) yields a plot of 1 + G(s) in the complex plane. By the Argument Principle, the number of clock- wise encirclements of the origin must be the number of zeros of 1 + G(s) in the right-half complex plane minus the poles of 1 + G(s) in the right-half complex plane. If instead, the contour is mapped through the open-loop transfer function G(s) , the result is the Nyquist plot of G(s) . By counting the resulting contour's encirclements of -1 , we find the difference between the number of poles and zeros in the right-half complex plane of 1 + G(s) . Recalling that the zeros of 1 + G(s) are the poles of the closed-loop system, and noting that the poles of 1 + G(s) are same as the poles of G(s) , we now state The Nyquist Criterion: Given a Nyquist contour r s , let P be the number of poles of G(s) encircled by r s and Z be the number of zeros of 1 + G(s) encircled by r s . Alternatively, and more importantly, Z is the number of poles of the closed loop system in the right half plane. The resultant contour in the G(s) -plane, 7" G , s x shall encircle (clock-wise) the point (-l + ;0) N times such that N = Z-P . For stability of a system, we must have Z = , i.e. the number of closed loop poles in the right half of the s-plane must be zero. Hence, the number of counterclockwise encirclements about (-1 + ;0) must be equal to P , the number of open loop poles in the right half plane (Faulkner, 1969), ( Franklin, 2002). 4.2 Bode diagram A Bode plot is a plot of either the magnitude or the phase of a transfer function T(jco) as a function of co . The magnitude plot is the more common plot because it represents the gain of the system. Therefore, the term "Bode plot" usually refers to the magnitude plot (Thomas, 2004), ( William, 1996), ( Willy, 2006). The rules for making Bode plots can be derived from the following transfer function: T(s) = K s V^oy where n is a positive integer. For +n as the exponent, the function has n zeros at s = . For -n, it has ft poles at s = 0. With s = jco, it follows that T(jco) = Kj ±n (co / co ) ±n , \T(jco)\ = Kj(co / co ) ±n and ZT(jco) = ±n x 90° . If co is increased by a factor of 10, |T(;&>)| changes by a factor of 10 ±n . Thus a plot of |T(;&>)| versus co on log- log scales has a slope of log(10 ±n ) = ±ft decades / decade . There are 20dBs in a decade , so the slope can also be expressed as ±20ft dB / decade . In order to give an example, (Fig. 5) shows the Bode diagrams of the first order and second order lag. Initial and final values of the phase lag courses can be seen clearly. The same holds for the initial values of the gain courses. Zero, the final value of these courses, lies at negative infinity, because of the logarithmic representation. Furthermore, for the second order lag the resonance magnification for smaller dampings can be see at the resonance frequency co . Even with a transfer function being given, a graphical analysis using these two diagrams might be clearer, and of course it can be tested more easily than, for example, a numerical analysis done by a computer. It will almost always be easier to estimate the effects of changes in the values of the parameters of the system, if we use a graphical approach instead of a numerical one. For this reason, today every control design software tool provides the possibility of computing the Nyquist plot or the Bode diagram for a given transfer function by merely clicking on a button. 10 Robust Control, Theory and Applications Bode Diagra First-order systems Bode Diagram i rrrnn i — r r rum 1 — r r rnrn 1 — r t trim r - I — rrrirr I I I I I I I I I I I ! ! J I I I I I I I I I I I I I I I I I I I I ' r ^^l~ : ^^! i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 — rrrnrn i — n i i i i i i i u r t r n t it 1 — i — r t titit r ~ 1 — r t r i t r J LLL1J1IJ J I L J_ l_l _L1 _L -^^^LT 1LI1I1 I J L _L J_ I _L I _L 1_L11LI1L i i i i i i i i i i i i i i i i i i i\/T\M. i j i r-^Jj^^timi i i i i mi 1 1- J- H-l 4-1 -1 \ 1 — -1— -1— 1 — 1 -1— 1 -1 4- - 1- 4- 4- UI4- l-l l — | l 4. Qnnn; -f— !— HI III Frequency (rad/seo) Second-order systems Fig. 5. Bode diagram of first and second-order systems 4.3 Evans root locus In addition to determining the stability of the system, the root locus can be used to design for the damping ratio and natural frequency of a feedback system (Franklin et aL, 2002). Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arcs whose center points coincide with the origin (see. Fig. 6). By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency a gain, K, can be calculated and implemented in the controller. More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead, PI, PD and PID controllers can be designed approximately with this technique. The definition of the damping ratio and natural frequency presumes that the overall feedback system is well approximated by a second order system, that is, the system has a dominant pair of poles. This often doesn't happen and so it's good practice to simulate the final design to check if the project goals are satisfied. Introduction to Robust Control Techniques 11 Root Locus -1 -0.5 Real Axis Fig. 6. Evans root locus of a second-order system Suppose there is a plant (process) with a transfer function expression P(s) , and a forward controller with both an adjustable gainK and a transfer function expression C(s) . A unity feedback loop is constructed to complete this feedback system. For this system, the overall transfer function is given by: T(8) K.C(s).P(s) l + K.C(s).P(s) (3) Thus the closed-loop poles of the transfer function are the solutions to the equation l + K.C(s).P(s) = . The principal feature of this equation is that roots may be found wherever K.C.P = -1 . The variability of K , the gain for the controller, removes amplitude from the equation, meaning the complex valued evaluation of the polynomial in s C(s).P(s) needs to have net phase of 180 deg, wherever there is a closed loop pole. The geometrical construction adds angle contributions from the vectors extending from each of the poles of KC to a prospective closed loop root (pole) and subtracts the angle contributions from similar vectors extending from the zeros, requiring the sum be 180. The vector formulation arises from the fact that each polynomial term in the factored CP , (s - a) for example, represents the vector from a which is one of the roots, to s which is the prospective closed loop pole we are seeking. Thus the entire polynomial is the product of these terms, and according to vector mathematics the angles add (or subtract, for terms in the denominator) and lengths multiply (or divide). So to test a point for inclusion on the root locus, all you do is add the angles to all the open loop poles and zeros. Indeed a form of protractor, the "spirule" was once used to draw exact root loci. From the function T(s) , we can also see that the zeros of the open loop system ( CP ) are also the zeros of the closed loop system. It is important to note that the root locus only gives the location of closed loop poles as the gain K is varied, given the open loop transfer function. The zeros of a system cannot be moved. 12 Robust Control, Theory and Applications Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of K varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of K. 5. Ingredients for a robust control The design of a control consists in adjusting the transfer function of the compensator so as to obtain the properties and the behavior wished in closed loop. In addition to the constraint of stability, we look typically the best possible performance. This task is complicated by two principal difficulties. On the one hand, the design is carried out on a idealized model of the system. We must therefore ensure the robustness to imperfections in the model, i.e. to ensure that the desired properties for a family of systems around the reference model. On the other hand, it faces inherent limitations like the compromise between performances and robustness. This section shows how these objectives and constraints can be formulated and quantified in a consistent framework favorable to their taking into systematic account. 5.1 Robustness to uncertainty The design of a control is carried out starting from a model of the real system often called nominal model or reference model. This model may come from the equations of physics or a process identification. In any case, this model is only one approximation of reality. Its deficiencies can be multiple: dynamic nonlinear ities neglected, uncertainty on certain physical parameters, assumptions simplifying, errors of measurement to the identification, etc.. In addition, some system parameters can vary significantly with time or operating conditions. Finally, from the unforeseeable external factors can come to disturb the operation of the control system. It is thus insufficient to optimize control compared to the nominal model: it is also necessary to be guarded against the uncertainty of modeling and external risks. Although these factors are poorly known, one has information in general on their maximum amplitude or their statistical nature. For example, the frequency of the oscillation, maximum intensity of the wind, or the terminals min and max on the parameter value. It is from this basic knowledge that one will try to carry out a robust control. There are two classes of uncertain factors. A first class includes the uncertainty and external disturbances. These are signals or actions randomness that disrupt the controlled system. They are identified according to their point of entry into the loop. Referring again to (Fig. 2) there are basically: • the disruption of the control w i which can come from errors of discretization or quantification of the control or parasitic actions on the actuators. • Disturbances at exit w corresponding to external effects on the output or unpredictable on the system, e.g. the wind for a airplane, an air pressure change for a chemical reactor, etc.. It should be noted that these external actions do not modify the dynamic behavior interns system, but only the "trajectory" of its outputs. A second class of uncertain factors joins together imperfections and variations of the dynamic model of the system. Recall that the robust control techniques applied to finite dimensional linear models, while real systems are generally non-linear and infinite Introduction to Robust Control Techniques 13 dimensional. Typically, the model used thus neglects non-linear ties and is valid only in one limited frequency band. It depends of more than physical parameters whose value can fluctuate and is often known only roughly. For practical reasons, one will distinguish: • the dynamic uncertainty which gathers the dynamic ones neglected in the model. There is usually only an upper bound on the amplitude of these dynamics. One must thus assume and guard oneself against worst case in the limit of this marker. • the parametric uncertainty or structured which is related to the variations or errors in estimation on certain physical parameters of the system, or with uncertainties of dynamic nature, but entering the loop at different points. Parametric uncertainty intervenes mainly when the model is obtained starting from the equations of physics. The way in which the parameters influential on the behavior of the system determines the "structure" of the uncertainty. 5.2 Representation of the modeling uncertainty The dynamic uncertainty (unstructured) can encompass of physical phenomena very diverse (linear or nonlinear, static or time-variant, frictions, hysteresis, etc.). The techniques discussed in this chapter are particularly relevant when one does not have any specific information if not an estimate of the maximum amplitude of dynamic uncertainty, In other words, when uncertainty is reasonably modeled by a ball in the space of bounded operators of £ 2 in 1 2 . Such a model is of course very rough and tends to include configurations with physical sense. If the real system does not comprise important nonlinearities, it is often preferable to be restricted with a stationary purely linear model of dynamic uncertainty. We can then balance the degree of uncertainty according to the frequency and translate the fact that the system is better known into low than in high frequency. Uncertainty is then represented as a disturbing system LTI AG(s) which is added to the nominal model G(s) of the real system: G true (s) = G(s) + AG(s) (4) This system must be BIBO-stable (bounded £ 2 in £ 2 ), and it usually has an estimate of the maximum amplitude of AG(jco) in each frequency band. Typically, this amplitude is small at lower frequencies and grows rapidly in the high frequencies where the dynamics neglected become important. This profile is illustrated in (Fig. 7). It defines a family of systems whose envelope on the Nyquist diagram is shown in (Fig. 8) (case SISO). The radius of the disk of the frequency uncertainty co is |zlG(;&>)| . IAGOH co Fig. 7. Standard profile for |zlG(;&>)| . 14 Robust Control, Theory and Applications Im(G(j©)) Re(GC/«D)) Fig. 8. Family of systems The information on the amplitude |zlG(;&>)| of the uncertainty can be quantified in several ways: • additive uncertainty: the real system is of the form: G tme (s) = G(s) + A(s) Where A(s) is a stable transfer function satisfying: ||W,(a>)2l(7'a>)W r (a>)|| <1 (5) (6) for certain models W^s) and W r (s) . These weighting matrices make it possible to incorporate information on the frequential dependence and directional of the maximum amplitude of A(s) (see singular values). • multiplicative uncertainty at the input: the real system is of the form: G true (s) = G(s).(I + A(s)) (7) where A(s) is like above. This representation models errors or fluctuations on the behavior in input. • multiplicative uncertainty at output: the real system is of the form: G tme (s) = (I + A(s)).G(s) (8) This representation is adapted to modeling of the errors or fluctuations in the output behavior. According to the data on the imperfections of the model, one will choose one or the other of these representations. Let us note that multiplicative uncertainty has a relative character. 5.3 Robust stability Let the linear system be given by the transfer function G(s) _ °m S + °m-l S a n s + a n _\S +... + a a s + aQ n( s ». T -e" =V+ ri(- +i ) where m < n (9) v=l v=l rv Introduction to Robust Control Techniques 1 5 with the gain V = ^- (10) First we must explain what we mean by stability of a system. Several possibilities exist to define the term, two of which we will discuss now. A third definition by the Russian mathematician Lyapunov will be presented later. The first definition is based on the step response of the system: Definition 1 A system is said to be stable if,fort->co, its step response converges to a finite value. Otherwise, it is said to be instable. This unit step function has been chosen to stimulate the system does not cause any restrictions, because if the height of the step is modified by the factor k, the values to the system output will change by the same factor k, too, according to the linearity of the system. Convergence towards a finite value is therefore preserved. A motivation for this definition can be the idea of following illustration: If a system converges towards a finished value after strong stimulation that a step in the input signal represents, it can suppose that it will not be wedged in permanent oscillations for other kinds of stimulations. It is obvious to note that according to this definition the first order and second order lag is stable, and that the integrator is instable. Another definition is attentive to the possibility that the input quantity may be subject to permanent changes: Definition 2 A linear system is called stable if for an input signal with limited amplitude, its output signal will also show a limited amplitude. This is the BIB O -Stability (bounded input - bounded output). Immediately, the question on the connection between the two definitions arises, that we will now examine briefly. The starting point of discussion is the convolution integral, which gives the relationship between the system's input and the output quantity (the impulse response): t t ¥(*) = j g(t~ T)x(r)dr = j* g(r)x(t - r)dr (11) x(t) is bounded if and only if \x(t)\ < k holds (with k> ) for all t . This implies: |y(')|* \\g(T)\\x(t-T)\dr<k\ g(T)dr (12) Now, with absolute convergence of the integral of the impulse response, Jl*(* \dr = c<oo (13) y(s) will be limited by kc , also, and thus the whole system will be BIBO-stable. Similarly it can be shown that the integral (13) converges absolutely for all BIBO-stable systems. BIBO 1 6 Robust Control, Theory and Applications stability and the absolute convergence of the impulse response integral are the equivalent properties of system. Now we must find the conditions under which the system will be stable in the sense of a finite step response (Definition 2): Regarding the step response of a system in the frequency domain, y(s) = G(s)i (14) S If we interpret the factor y as an integration (instead of the Laplace transform of the step signal), we obtain y(s)= J g(r)dT (15) in the time domain for y(0) = 0.y(t) converge to a finite value only if the integral converges: = C < 00 (16) J g(T)dr Convergence is obviously a weaker criterion than absolute convergence. Therefore, each BIBO-stable system will have a finite step response. To treat the stability always in the sense of the BIBO-stability is tempting because this stronger definition makes other differentiations useless. On the other hand, we can simplify the following considerations much if we use the finite-step-response-based definition of stability (Christopher, 2005), (Arnold, 2006). In addition to this, the two definitions are equivalent as regards the transfer functions anyway. Consequently, henceforth we will think of stability as characterized in (Definition 2). Sometimes stability is also defined while requiring that the impulse response to converge towards zero for t — » oo . A glance at the integral (16) shows that this criterion is necessary but not sufficient condition for stability as defined by (Definition 2), while (Definition 2) is the stronger definition. If we can prove a finite step response, then the impulse response will certainly converge to zero. 5.3.1 Stability of a transfer function If we want to avoid having to explicitly calculate the step response of a system in order to prove its stability, then a direct examination of the transfer function of the system's, trying to determine criteria for the stability, seems to suggest itself ( Levine, 1996). This is relatively easy concerning all ideas that we developed up to now about the step response of a rational transfer function. The following theorem is valid: Theorem 2 A transfer element with a rational transfer function is stable in the sense of (Definition 2) if and only if all poles of the transfer function have a negative real part. According to equation (17), the step response of a rational transfer element is given by: y(*) = ZM0« 8 * (17) Introduction to Robust Control Techniques 17 For each pole s A of multiplicity n A , we obtain a corresponding operand h A (t)e A * , which h A (t) is a polynomial of degree n x - 1 . For a pole with a negative real part, this summand disappears to increase t, as the exponential function converges more quickly towards zero than the polynomial h x (t) can increase. If all the poles of the transfer function have a negative real part, then all corresponding terms disappear. Only the summand h { (t)e [i for the simple pole s i = remains, due to the step function. The polynomial h { (t) is of degree n { - 1 = , i.e. a constant, and the exponential function is also reduced to a constant. In this way, this summand form the finite final value of the step function, and the system is stable. We omit the proof in the opposite direction, i.e. a system is instable if at least one pole has a positive real part because it would not lead to further insights. It is interesting that (Theorem 2) holds as well for systems with delay according to (9). The proof of this last statement will be also omitted. Generally, the form of the initial transients as reaction to the excitations of outside will also be of interest besides that the fact of stability. If a plant has, among others, a complex conjugate pair of poles s x , s x , the ratio |Re(s A )| ^Re(s A ) 2 +Im(s /l ) 2 is equal to the damping ratio D and therefore responsible for the form of the initial transient corresponding to this pair of poles. In practical applications one will therefore pay attention not only to that the system's poles have a negative real part, but also to the damping ratio D having a sufficiently high value, i.e. that a complex conjugate pair of poles lies at a reasonable distance to the axis of imaginaries. 5.3.2 Stability of a control loop The system whose stability must be determined will in the majority of the cases be a closed control loop (Goodwin, 2001), as shown in (Fig. 2). A simplified structure is given in (Fig. 9). Let the transfer function of the control unit is K(s) , the plant will be given by G(s) and the metering element by M(s) . To keep further derivations simple, we set M(s) to 1, i.e. we neglect the dynamic behavior of the metering element, for simple cases, but it should normally be no problem to take the metering element also into consideration. CO •9 K-^-> -► K M <*- y Fig. 9. Closed-loop system We summarize the disturbances that could affect the closed loop system to virtually any point, into a single disturbance load that we impressed at the plant input. This step simplifies the theory without the situation for the controller easier than it would be in practical applications. Choose the plant input as the point where the disturbance affects the plant is most unfavorable: The disturbance can affect plants and no countermeasure can be applied, as the controller can only counteract after the changes at the system output. 18 Robust Control, Theory and Applications To be able to apply the criteria of stability to this system we must first calculate the transfer function that describes the transfer characteristic of the entire system between the input quantity co and the output quantity y . This is the transfer function of the closed loop, which is sometimes called the reference (signal) transfer function. To calculate it, we first set d to zero. In the frequency domain we get y(s) = G(s)u(s) = G(s)K(s)(a>(s) - y(s)) (18) ty x y( s ) G ( S ) K ( S ) T(s) = *±± = w w H9) w a)(s) G(s)K(s) + l { } In a similar way, we can calculate a disturbance transfer function, which describes the transfer characteristic between the disturbance d and the output quantity y: c ^ V( s ) G ( S ) K ( S ) S(s) = ^-^ = — — (20) w d(s) G(s)K(s) + l { } The term G(s)K(s) has a special meaning: if we remove the feedback loop, so this term represents the transfer function of the resulting open circuit. Consequently, G(s)K(s) is sometimes called the open-loop transfer function. The gain of this function (see (9)) is called open-loop gain. We can see that the reference transfer function and the disturbance transfer function have the same denominator G(s)K(s) + 1 . On the other hand, by (Theorem 2), it is the denominator of the transfer function that determines the stability. It follows that only the open-loop transfer function affects the stability of a system, but not the point of application of an input quantity. We can therefore restrict an analysis of the stability to a consideration of the termG(s)K(s) + l. However, since both the numerator and denominator of the two transfer functions T(s) and S(s) are obviously relatively prime to each other, the zeros of G(s)K(s) + l are the poles of these functions, and as a direct consequence of (Theorem 2) we can state: Theorem 3 A closed-loop system with the open-loop transfer function G(s)K(s) is stable if and only if all solutions of the characteristic equation have a negative real part. G(s)K(s) + 1 = (21) Computing these zeros in an analytic way will no longer be possible if the degree of the plant is greater than two, or if an exponential function forms a part of the open-loop transfer function. Exact positions of the zeros, though, are not necessary in the analysis of stability. Only the fact whether the solutions have a positive or negative real part is of importance. For this reason, in the history of the control theory criteria of stability have been developed that could be used to determine precisely without having to make complicated calculations (Christopher, 2005), ( Franklin, 2002). 5.3.3 Lyapunov's stability theorem We state below a variant of Lyapunov's direct method that establishes global asymptotic stability. Introduction to Robust Control Techniques 19 Theorem 4 Consider the dynamical system x(t) = f(x(t)) and let x = 0be its unique equilibrium point. If there exists a continuously differentiable function V : 9? n — » 9? such that V(0) = (22) V(x)yO Vx^O (23) > y(x) -» oo (24) V(x)^0 Vx*0, (25) then x = is globally asymptotically stable. Condition (25) is what we refer to as the monotonicity requirement of Lyapunov's theorem. In the condition, V(x) denotes the derivation of V(x) along the trajectories of x(t) and is given by V(x) dV(x) .. . where <.,.> denotes the standard inner product in 9? n and dV(x) dx g ^H n is the gradient of V(x) . As far as the first two conditions are concerned, it is only needed to assume that V(x) is lower bounded and achieves its global minimum at x = .There is no conservatism, however, in requiring (22) and (23). A function satisfying condition (24) is called radially unbounded. We refer the reader to (Khalil, 1992) for a formal proof of this theorem and for an example that shows condition (24) cannot be removed. Here, we give the geometric intuition of Lyapunov's theorem, which essentially carries all of the ideas behind the proof. Fig. 10. Geometric interpretation of Lyapunov's theorem. (Fig. 10) shows a hypothetical dynamical system in $R 2 . The trajectory is moving in the (x 1/ x 2 ) plane but we have no knowledge of where the trajectory is as a function of time. On the other hand, we have a scalar valued function V(x) , plotted on the z-axis, which has the 20 Robust Control, Theory and Applications guaranteed property that as the trajectory moves the value of this function along the trajectories strictly decreases. Since V(x(t)) is lower bounded by zero and is strictly decreasing, it must converge to a nonnegative limit as time goes to infinity. It takes a relatively straightforward argument appealing to continuity of V(x) and V(x) ) to show that the limit of V(x(t)) cannot be strictly positive and indeed conditions (22)-(25) imply V(x(t)) -> as t -> oo Since x = is the only point in space where V(x) vanishes, we can conclude that x(t) goes to the origin as time goes to infinity. It is also insightful to think about the geometry in the (x 1 ,x 2 ) plane. The level sets of V(x) are plotted in (Fig. 10) with dashed lines. Since V(x(t )) decreases monotonically along trajectories, we can conclude that once a trajectory enters one of the level sets, say given by V (x) = c , it can never leave the set q •= l x e 9? w \V < c) -This property is known as invariance of sub-level sets. Once again we emphasize that the significance of Lyapunov' s theorem is that it allows stability of the system to be verified without explicitly solving the differential equation. Lyapunov' s theorem, in effect, turns the question of determining stability into a search for a so-called Lyapunov function, a positive definite function of the state that decreases monotonically along trajectories. There are two natural questions that immediately arise. First, do we even know that Lyapunov functions always exist? Second, if they do in fact exist, how would one go about finding one? In many situations, the answer to the first question is positive. The type of theorems that prove existence of Lyapunov functions for every stable system are called converse theorems. One of the well known converse theorems is a theorem due to Kurzweil that states if / in (Theorem 4) is continuous and the origin is globally asymptotically stable, then there exists an infinitely differentiable Lyapunov function satisfying conditions of (Theorem 4). We refer the reader to (Khalil, 1992) and (Bacciotti & Rosier,2005) for more details on converse theorems. Unfortunately, converse theorems are often proven by assuming knowledge of the solutions of (Theorem 4) and are therefore useless in practice. By this we mean that they offer no systematic way of finding the Lyapunov function. Moreover, little is known about the connection of the dynamics /to the Lyapunov function V. Among the few results in this direction, the case of linear systems is well settled since a stable linear system always admits a quadratic Lyapunov function. It is also known that stable and smooth homogeneous systems always have a homogeneous Lyapunov function (Rosier, 1992). 5.3.4 Criterion of Cremer, Leonhard and Michailow Initially let us discuss a criterion which was developed independently by Cremer , Leonhard and Michailov during the years 1938-1947. The focus of interest is the phase shift of the Nyquist plot of a polynomial with respect to the zeros of the polynomial (Mansour, 1992). Consider a polynomial of the form n P(s) = s n + a n _ x s n ~ x + ... + a x s + a = Y[(s - s v ) (26) Introduction to Robust Control Techniques 21 be given. Setting s = jco and substituting we obtain njco) = f[{jco-s v ) = fl{\jm-s v \)e^ « ;2>,.(») . =ni^- s vi e ^ =i p o' (27) j<p(co) We can see, that the frequency response P(jco) is the product of the vectors (jco - s v ) , where the phase <£>(&>) is given by the sum of the angles (p v (co) of those vectors. (Fig.ll) shows the situation corresponding to a pair of complex conjugated zeros with negative real part and one zero with a positive real part. (yco-^) *3 Re Fig. 11. Illustration to the Cremer-Leonhard-Michailow criterion If the parameter co traverses the interval (-00,00), it causes the end point of the vectors (jo)-s v ) to move along the axis of imaginaries in positive direction. For zeros with negative real part, the corresponding angle (p v traverses the interval from — to +— , for zeros with 2 2 positive real part the interval from -\ — — to +— . For zeros lying on the axis of imaginaries 2 2 .— and switches to the value +— 2 2 the corresponding angle cp v initially has the value at jco = s v . We will now analyze the phase of frequency response, i.e. the entire course which the angle (p(co) takes. This angle is just the sum of the angles (pu v (co) . Consequently, each zero with a negative real part contributes an angle of +n to the phase shift of the frequency response, and each zero with a positive real part of the angle -n . Nothing can be said about zeros located on the imaginary axis because of the discontinuous course where the values of the phase to take. But we can immediately decide zeros or not there watching the Nyquist plot of the polynomial P(s) . If she got a zero purely imaginary s = s v , the corresponding Nyquist plot should pass through the origin to the frequency &> = |s v | . This leads to the following theorem: 22 Robust Control, Theory and Applications Theorem 5 A polynomial P(s) of degree n with real coefficients will have only zeros with negative real part if and only if the corresponding Nyquist plot does not pass through the origin of the complex plane and the phase shift A<p of the frequency response is equal to nn for -go < co < +00 . If a traverses the interval < co< +00 only, then the phase shift needed will he equal to V_ n . 2 We can easily prove the fact that for < co< +00 the phase shift needed is only — n —only 2 half the value: For zeros lying on the axis of reals, it is obvious that their contribution to the phase shift will be only half as much if co traverses only half of the axis of imaginaries (from to 00 ). The zeros with an imaginary part different from zero are more interesting. Because of the polynomial's real-valued coefficients, they can only_appear as a pair of complex conjugated zeros. (Fig. 12) shows such a pair with s 1 = s 2 and a x = -a 2 . For -00 <co< +00 the contribution to the phase shift by this pair is 2n . For < co< +00 , the contribution of s 1 is — + \a J and the one for s 2 is Ul. Therefore, the overall contribution of this pair of 2 ' ' 2 ' ' poles is n , so also for this case the phase shift is reduced by one half if only the half axis of imaginaries is taken into consideration. Fig. 12. Illustration to the phase shift for a complex conjugated pair of poles 6. Beyond this introduction There are many good textbooks on Classical Robust Control. Two popular examples are (Dorf & Bishop, 2004) and (Franklin et al., 2002). A less typical and interesting alternative is the recent textbook (Goodwin et al., 2000). All three of these books have at least one chapter devoted to the Fundamentals of Control Theory. Textbooks devoted to Robust and Optimal Control are less common, but there are some available. The best known is probably (Zhou et al.1995). Other possibilities are (Astrom & Wittenmark, 1996), (Robert, 1994)( Joseph et al, 2004). An excellent book about the Theory and Design of Classical Control is the one by Astrom and Hagglund (Astrom & Hagglund, 1995). Good references on the limitations of control are (Looze & Freudenberg, 1988). Bode r s book (Bode, 1975) is still interesting, although the emphasis is on vacuum tube circuits. Introduction to Robust Control Techniques 23 7. References Astrom, K. J. & Hagglund, T. (1995). PID Controllers: Theory, Design and Tuning, International Society for Measurement and Control, Seattle, WA, , 343p, 2nd edition. ISBN: 1556175167. Astrom, K. J. & Wittenmark, B. (1996). Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, NJ, 555p, 3rd edition. ISBN-10: 0133148998. Arnold Zankl (2006). Milestones in Automation: From the Transistor to the Digital Factory, Wiley-VCH, ISBN 3-89578-259-9. Bacciotti, A. & Rosier, L. (2005). Liapunov functions and stability in control theory, Springer, 238 p, ISBN:3540213325. Bode, H. W. (1975). Network Analysis and Feedback Amplifier Design, R. E. Krieger Pub. Co., Huntington, NY. Publisher: R. E. Krieger Pub. Co; 577p, 14th print. ISBN 0882752421. Christopher Kilian (2005). Modern Control Technology. Thompson Delmar Learning, ISBN 1- 4018-5806-6. Dorf, R. C. & Bishop, R. H. (2005). Modern Control Systems, Prentice-Hall, Upper Saddle River, NJ, 10* edition. ISBN 0131277650. Faulkner, E.A. (1969): Introduction to the Theory of Linear Systems, Chapman & Hall; ISBN 0- 412-09400-2. Franklin, G. F.; Powell, J. D. & Emami-Naeini, A. (2002). Feedback Control of Dynamical Systems, Prentice-Hall, Upper Saddle River, NJ, 912p, 4th edition. ISBN: 0-13- 032393-4. Joseph L. Hellerstein; Dawn M. Tilbury, & Sujay Parekh (2004). Feedback Control of Computing Systems, John Wiley and Sons. ISBN 978-0-471-26637-2. Boukhetala, D.; Halbaoui, K. and Boudjema, F.(2006). Design and Implementation of a Self tuning Adaptive Controller for Induction Motor Drives. International Review of Electrical Engineering, 260-269, ISSN: 1827- 6660. Goodwin, G. C, Graebe, S. F. & Salgado, M. E. (2000). Control System Design, Prentice-Hall, Upper Saddle River, NJ.,908p, ISBN: 0139586539. Goodwin, Graham .(2001). Control System Design, Prentice Hall. ISBN 0-13-958653-9. Khalil, H. (2002). Nonlinear systems, Prentice Hall, New Jersey, 3rd edition. ISBN 0130673897. Looze, D. P & Freudenberg, J. S. (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems, Springer- Verlag, Berlin. , 281p, ISBN:038718869. Levine, William S., ed (1996). The Control Handbook, New York: CRC Press. ISBN 978-0-849- 38570-4. Mansour, M. (1992). The principle of the argument and its application to the stability and robust stability problems, Springer Berlin - Heidelberg, Vo 183, 16-23, ISSN 0170-8643, ISBN 978-3-540-55961-0. Robert F. Stengel (1994). Optimal Control and Estimation, Dover Publications. ISBN 0-486- 68200-5. Rosier, L. (1992). Homogeneous Lyapunov function for homogeneous continuous vector field, Systems Control Lett, 19(6):467-473. ISSN 0167-6911 Thomas H. Lee (2004). The design of CMOS radio- frequency integrated circuits, (Second Edition ed.). Cambridge UK: Cambridge University Press, p. §14.6 pp. 451-453.ISBN 0-521- 83539-9. 24 Robust Control, Theory and Applications Zhou, K., Doyle J. C, & Glover, K.. (1995). Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ., 596p, ISBN: 0134565673. William S Levine (1996). The control handbook: the electrical engineering handbook series, (Second Edition ed.). Boca Raton FL: CRC Press/IEEE Press, p. §10.1 p. 163. ISBN 0849385709. Willy M C Sansen (2006) .Analog design essentials, Dordrecht, The Netherlands: Springer. p. §0517-§0527 pp. 157-163.ISBN 0-387-25746-2. Robust Control of Hybrid Systems Khaled Halbaoui 1 ' 2 , Djamel Boukhetala 2 and Fares Boudjema 2 tPower Electronics Laboratory, Nuclear Research Centre of Brine CRNB, BP 180 Ain oussera 17200, Djelfa, 2 Laboratoire de Commande des Processus, ENSP, 10 avenue Pasteur, Hassan Badi, BP 182 El-Harrach, Algeria 1. Introduction The term "hybrid systems" was first used in 1966 Witsenhausen introduced a hybrid model consisting of continuous dynamics with a few sets of transition. These systems provide both continuous and discrete dynamics have proven to be a useful mathematical model for various physical phenomena and engineering systems. A typical example is a chemical batch plant where a computer is used to monitor complex sequences of chemical reactions, each of which is modeled as a continuous process. In addition to the discontinuities introduced by the computer, most physical processes admit components (eg switches) and phenomena (eg collision), the most useful models are discrete. The hybrid system models arise in many applications, such as chemical process control, avionics, robotics, automobiles, manufacturing, and more recently molecular biology. The control design for hybrid systems is generally complex and difficult. In literature, different design approaches are presented for different classes of hybrid systems, and different control objectives. For example, when the control objective is concerned with issues such as safety specification, verification and access, the ideas in discrete event control and automaton framework are used for the synthesis of control. One of the most important control objectives is the problem of stabilization. Stability in the continuous systems or not-hybrid can be concluded starting from the characteristics from their fields from vectors. However, in the hybrid systems the properties of stability also depend on the rules of commutation. For example, in a hybrid system by commutation between two dynamic stable it is possible to obtain instabilities while the change between two unstable subsystems could have like consequence stability. The majority of the results of stability for the hybrid systems are extensions of the theories of Lyapunov developed for the continuous systems. They require the Lyapunov function at consecutive switching times to be a decreasing sequence. Such a requirement in general is difficult to check without calculating the solution of the hybrid dynamics, and thus losing the advantage of the approach of Lyapunov. In this chapter, we develop tools for the systematic analysis and robust design of hybrid systems, with emphasis on systems that require control algorithms, that is, hybrid control systems. To this end, we identify mild conditions that hybrid equations need to satisfy so that their behavior captures the effect of arbitrarily small perturbations. This leads to new concepts of global solutions that provide a deep understanding not only on the robustness 26 Robust Control, Theory and Applications properties of hybrid systems, but also on the structural properties of their solutions. Alternatively, these conditions allow us to produce various tools for hybrid systems that resemble those in the stability theory of classical dynamical systems. These include general versions of theorems of Lyapunov stability and the principles of invariance of LaSalle. 2. Hybrid systems: Definition and examples Different models of hybrid systems have been proposed in the literature. They mainly differ in the way either the continuous part or the discrete part of the dynamics is emphasized, which depends on the type of systems and problems we consider. A general and commonly used model of hybrid systems is the hybrid automaton (see e.g. (Dang, 2000) and (Girard, 2006)). It is basically a finite state machine where each state is associated to a continuous system. In this model, the continuous evolutions and the discrete behaviors can be considered of equal complexity and importance. By combining the definition of the continuous system, and discrete event systems hybrid dynamical systems can be defined: Definition 1 A hybrid system H is a collection H := (Q,X,Z ,U,F ,R) , where Q is a finite set, called the set of discrete states; X cz 9? n is the set of continuous states; I is a set of discrete input events or symbols; X cz $i m is the set of continuous inputs; F : Q x X x LI — » $i n is a vector field describing the continuous dynamics; R:QxXxi7xlI^QxX describes the discrete dynamics. off[ 3 Time Fig. 1. A trajectory of the room temperature. Example 1 (Thermostat). The thermostat consists of a heater and a thermometer which maintain the temperature of the room in some desired temperature range (Rajeev, 1993). The lower and upper thresholds of the thermostat system are set at x m and x M such that x m <x M . The heater is maintained on as long as the room temperature is below x M , and it is turned off whenever the thermometer detects that the temperature reaches x M . Similarly, the heater remains off if the temperature is above x m and is switched on whenever the Robust Control of Hybrid Systems 27 temperature falls to x m (Fig. 1). In practical situations, exact threshold detection is impossible due to sensor imprecision. Also, the reaction time of the on/ off switch is usually non-zero. The effect of these inaccuracies is that we cannot guarantee switching exactly at the nominal values x m and x M . As we will see, this causes non-determinism in the discrete evolution of the temperature. Formally we can model the thermostat as a hybrid automaton shown in (Fig. 2). The two operation modes of the thermostat are represented by two locations W and 'off. The on/off switch is modeled by two discrete transitions between the locations. The continuous variable x models the temperature, which evolves according to the following equations. Fig. 2. Model of the thermostat. • If the thermostat is on, the evolution of the temperature is described by: x = f x (x, u) = -x + 4 + u a) When the thermostat is off, the temperature evolves according to the following differential equation: x = f 2 (x,u) = -x + u xM xM-e xO xm+e :-"-: \ : xm e Fig. 3. Two different behaviors of the temperature starting at x . The second source of non-determinism comes from the continuous dynamics. The input signal u of the thermostat models the fluctuations in the outside temperature which we cannot control. (Fig. 3 left) shows this continuous non-determinism. Starting from the initial temperature x , the system can generate a "tube" of infinite number of possible trajectories, each of which corresponds to a different input signal u . To capture uncertainty of sensors, we define the first guard condition of the transition from W to 'off as an interval [x M -e,x M +s] with s y . This means that when the temperature enters this interval, the thermostat can either turn the heater off immediately or keep it on for some time provided 28 Robust Control, Theory and Applications that x < x M + s . (Fig. 3 right) illustrates this kind of non-determinism. Likewise, we define the second guard condition of the transition from ] off to W as the interval \x m -£,x m +s] . Notice that in the thermostat model, the temperature does not change at the switching points, and the reset maps are thus the identity functions. Finally we define the two staying conditions of the W and ] off locations as x< x M + s and x > x M - s respectively, meaning that the system can stay at a location while the corresponding staying conditions are satisfied. Example 2 (Bouncing Ball). Here, the ball (thought of as a point-mass) is dropped from an initial height and bounces off the ground, dissipating its energy with each bounce. The ball exhibits continuous dynamics between each bounce; however, as the ball impacts the ground, its velocity undergoes a discrete change modeled after an inelastic collision. A mathematical description of the bouncing ball follows. Let x x := h be the height of the ball and x 2 '= h (Fig. 4). A hybrid system describing the ball is as follows: £«:= -y.x 2 ,D: = 0, x 2 -< 0} f{x ,C-={x:x 1 >0}\D . (2) This model generates the sequence of hybrid arcs shown in (Fig. 5). However, it does not generate the hybrid arc to which this sequence of solutions converges since the origin does not belong to the jump set D . This situation can be remedied by including the origin in the jump set D . This amounts to replacing the jump set D by its closure. One can also replace the flow set C by its closure, although this has no effect on the solutions. It turns out that whenever the flow set and jump set are closed, the solutions of the corresponding hybrid system enjoy a useful compactness property: every locally eventually bounded sequence of solutions has a subsequence converging to a solution. h = 0&h~<0? !!!!!! [ i [ \ [ J A - h ;;;;•; 4« Fig. 4. Diagram for the bouncing ball system h \ K h \ -^ \ -h r \ \ \ N ^ \ \ N \ Fig. 5. Solutions to the bouncing ball system Consider the sequence of hybrid arcs depicted in (Fig. 5). They are solutions of a hybrid "bouncing ball" model showing the position of the ball when dropped for successively Robust Control of Hybrid Systems 29 lower heights, each time with zero velocity. The sequence of graphs created by these hybrid arcs converges to a graph of a hybrid arc with hybrid time domain given by {0} x fnonnegative integers/ where the value of the arc is zero everywhere on its domain. If this hybrid arc is a solution then the hybrid system is said to have a "compactness" property. This attribute for the solutions of hybrid systems is critical for robustness properties. It is the hybrid generalization of a property that automatically holds for continuous differential equations and difference equations, where nominal robustness of asymptotic stability is guaranteed. Solutions of hybrid systems are hybrid arcs that are generated in the following way: Let C and D be subsets of 9? n and let / , respectively g , be mappings from C , respectively D , to yi n . The hybrid system H := (f,g,C,D) can be written in the form x = f(x) xeC (3) x + = g(x) xgD w The map /is called the "flow map", the map g is called the "jump map", the set C is called the "flow set", and the set D is called the "jump set". The state x may contain variables taking values in a discrete set (logic variables), timers, etc. Consistent with such a situation is the possibility that C U D is a strict subset of 9? n . For simplicity, assume that / and g are continuous functions. At times it is useful to allow these functions to be set-valued mappings, which will denote by F and G , in which case F and G should have a closed graph and be locally bounded, and F should have convex values. In this case, we will write x eF x eC x + e G x e D (4) A solution to the hybrid system (4) starting at a point x e C u D is a hybrid arc x with the following properties: 1. x(0,0) = x ; 2. given (s,j) e dom x , if there exists x >- s such that (x,;) e dom x , then, for all t e [s,x] , x{t,]) g C and, for almost all t e [s,x] , x(t,j) e F(x(t,j)) ; 3. given (i, f) s dom x , if (t,j + 1) e dom x then x(t,j) e D and x(t,j + 1) e G(x(t,j)) . Solutions from a given initial condition are not necessarily unique, even if the flow map is a smooth function. 3. Approaches to analysis and design of hybrid control systems The analysis and design tools for hybrid systems in this section are in the form of Lyapunov stability theorems and LaSalle-like invariance principles. Systematic tools of this type are the base of the theory of systems for purely of the continuous-time and discrete-time systems. Some similar tools available for hybrid systems in (Michel, 1999) and (DeCarlo, 2000), the tools presented in this section generalize their conventional versions of continuous-time and discrete-time hybrid systems development by defining an equivalent concept of stability and provide extensions intuitive sufficient conditions of stability asymptotically. 30 Robust Control, Theory and Applications 3.1 LaSalle-like invariance principles Certain principles of invariance for the hybrid systems have been published in (Lygeros et al, 2003) and (Chellaboina et al., 2002). Both results require, among other things, unique solutions which is not generic for hybrid control systems. In (Sanfelice et al., 2005), the general invariance principles were established that do not require uniqueness. The work in (Sanfelice et al., 2005) contains several invariance results, some involving integrals of functions, as for systems of continuous-time in (Byrnes & Martin, 1995) or (Ryan, 1998), and some involving nonincreasing energy functions, as in work of LaSalle (LaSalle, 1967) or (LaSalle, 1976). Such a result will be described here. Suppose we can find a continuously differentiable function V : 9? n -> 9? such that u c (x) := (W(x),f(x)) < VigC u d (x) := V(g(x)) - V(x) < VigD (5) Consider x(-,-) a bounded solution with an unbounded hybrid time. Then there exists a value r in the range V so that x tends to the largest weakly invariant set inside the set M r := V-\r)n(u-\0)[){ui(p) g (uf(Q)j)) (6) where w^(0) : the set of points x satisfying u d (x) = and g(w^ a (0)) corresponds to the set of points g(y) where y e uf(0) . The naive combination of continuous-time and discrete-time results would omit the intersection withg(w^ a (0)) . This term, however, can be very useful for zeroing in set to which trajectories converge. 3.2 Lyapunov stability theorems Some preliminary results on the existence of the non-smooth Lyapunov function for the hybrid systems published in (DeCarlo, 2000). The first results on the existence of smooth Lyapunov functions, which are closely related to the robustness, published in (Cai et al., 2005). These results required open basins of attraction, but this requirement has since been relaxed in (Cai et al. 2007). The simplified discussion here is borrowed from this posterior work. Let be an open subset of the state space containing a given compact set A and let co : — » 9?> be a continuous function which is zero for all x e A , is positive otherwise, which grows without limit as its argument grows without limit or near the limit . Such a function is called a suitable indicator for the compact set A in the open set . An example of such a function is the standard function on 9? n which is an appropriate indicator of origin. More generally, the distance to a compact set A is an appropriate indicator for all A on 9? n . Given an open setc7, an appropriate indicator co and hybrid data(/,g,C,D) , a function V : — > 9?> is called a smooth Lyapunov function for (/ ,g,C,D,a>,G) if it is smooth and there exist functions a a ,a 2 belonging to the class- %^ , such as a a (co(x)) < V(x) < a 2 (co(x)) Vx e (VV(x),f(x))<-V(x) \/xeCf]0 (7) V(g(x)) < e~ x V (x) \/xeD{]0 Suppose that such a function exists, it is easy to verify that all solutions for the hybrid system (f,g,C,D) from fl (c U D\ satisfied Robust Control of Hybrid Systems 31 (o(x(t f j)) < 04 1 (e~ H a 2 (co(x(0,0)))) \/(t,j) e dom x (8) In particular, • (pre-stability of A ) for each e ^ there exists 5^0 such that x(0,0) e A + 5B implies, for each generalized solution, that x(t,j) e A + sB for all (£,;) e dom x , and • (before attractive AonO ) any generalized solution from fl (C U D ) is bounded and if its time domain is unbounded, so it converges to A . According to one of the principal results in (Cai et aL, 2006) there exists a smooth Lyapunov function for (/ ' ,g f C,D,<d,Q)if and only if the set A is pre-stable and pre-attractive on and is forward invariant (i.e., x(0,0) e fl (C U D ) implies x(f,;) e tf for all (£,;') g rfom x ). One of the primary interests in inverse Lyapunov theorems is that they can be employed to establish the robustness of the asymptotic stability of various types of perturbations. 4. Hybrid control application In system theory in the 60s researchers were discussing mathematical frameworks so to study systems with continuous and discrete dynamics. Current approaches to hybrid systems differ with respect to the emphasis on or the complexity of the continuous and discrete dynamics, and on whether they emphasize analysis and synthesis results or analysis only or simulation only. On one end of the spectrum there are approaches to hybrid systems that represent extensions of system theoretic ideas for systems (with continuous- valued variables and continuous time) that are described by ordinary differential equations to include discrete time and variables that exhibit jumps, or extend results to switching systems. Typically these approaches are able to deal with complex continuous dynamics. Their main emphasis has been on the stability of systems with discontinuities. On the other end of the spectrum there are approaches to hybrid systems embedded in computer science models and methods that represent extensions of verification methodologies from discrete systems to hybrid systems. Several approaches to robustness of asymptotic stability and synthesis of hybrid control systems are represented in this section. 4.1 Hybrid stabilization implies input-to-state stabilization In the paper (Sontag, 1989) it has been shown, for continuous-time control systems, that smooth stabilization involves smooth input-to-stat stabilization with respect to input additive disturbances. The proof was based on converse Lyapunov theorems for continuous-time systems. According to the indications of (Cai et aL, 2006), and (Cai et al. 2007), the result generalizes to hybrid control systems via the converse Lyapunov theorem. In particular, if we can find a hybrid controller, with the type of regularity used in sections 4.2 and 4.3, to achieve asymptotic stability, then the input-to-state stability with respect to input additive disturbance can also be achieved. Here, consider the special case where the hybrid controller is a logic-based controller where the variable takes values in the logic of a finite set. Consider the hybrid control system 9f:= k = m + r\ q mu q+ »d) ^eC q ,qeQ (9) 32 Robust Control, Theory and Applications where Q is a finite index set, for each q e Q , f , rj : C ^> 9? w are continuous functions, C and D are closed and G has a closed graph and is locally bounded. The signal u is the control, and d is the disturbance, while u- is vector that is independent of the state, input, and disturbance. Suppose 9£ is stabilizable by logic-based continuous feedback; that is, for the case where d = , there exist continuous functions k n defined on C n such that, with u := k q (Z) , the nonempty and compact set A = \J qA x |^| is pre-stable and globally pre- attractive. Converse Lyapunov theorems can then be used to establish the existence of a logic-based continuous feedback that renders the closed-loop system input-to-state stable with respect to d . The feedback has the form = *,G)-s.i£(!;)VV,(!;) (10) where e ^ and V (£) is a smooth Lyapunov function that follows from the assumed asymptotic stability when d = . There exist class- ^ functions 04 and oc 2 such that, with this feedback control, the following estimate holds: M. Mt,j) <max r 1 (2.exp(- f -;).a 2 (|^(0,0)|^ (00) )) / a^ 2.8 " "°° (11) where f^ :=sup {8fi)edomd \d(s,i 4.2 Control Lyapunov functions Although the control design using a continuously differentiable control-Lyapunov function is well established for input-affine nonlinear control systems, it is well known that not all controllable input-affine nonlinear control system function admits a continuously differentiable control-Lyapunov function. A well known example in the absence of this control-Lyapunov function is the so-called "Brockett", or "nonholonomic integrator". Although this system does not allow continuously differentiable control Lyapunov function, it has been established recently that admits a good "patchy" control-Lyapunov function. The concept of control-Lyapunov function, which was presented in (Goebel et al., 2009), is inspired not only by the classical control-Lyapunov function idea, but also by the approach to feedback stabilization based on patchy vector fields proposed in (Ancona & Bressan, 1999). The idea of control-Lyapunov function was designed to overcome a limitation of discontinuous feedbacks, such as those from patchy feedback, which is a lack of robustness to measurement noise. In (Goebel et al., 2009) it has been demonstrated that any asymptotically controllable nonlinear system admits a smooth patchy control-Lyapunov function if we admit the possibility that the number of patches may need to be infinite. In addition, it was shown how to construct a robust stabilizing hybrid feedback from a patchy control-Lyapunov function. Here the idea when the number of patches is finite is outlined and then specialized to the nonholonomic integrator. Generally , a global patchy smooth control-Lyapunov function for the origin for the control system x = f(x,u) in the case of a finite number of patches is a collection of functions V and sets Q and f2' where q e Q := { 1, . . . , m } , such as a. for each q e Q , Q and f2' q are open and . tf:=9r\{0} = U^=U, £Q ^ • for each q e Q , the outward unit normal to dQ q is continuous on ( df2 q \ U r ^ q 0' r J fl , Robust Control of Hybrid Systems 33 • for each;? e Q , f2' q f)0 a Q q ; b. for each q e Q , V is a smooth function defined on a neighborhood (relative to ) oiQ q . c. there exist a continuous positive definite function a and class- %^ functions y and y such that . y(|4<V„(x)<y(|*|) V^c,eQ, xe ^\[j^n' r )f]0; • for each q e Q and x e lfi\\J ry Q[ 1 there exists u x such that (W q (x),f(x,u x ,q))<-a(x) • for each q e Q and x e ( /2 \ U r ^ (? ^ ) fl there exists u x such that (v^(x),/(x, Wx ,^))<-a(x) ( n q ( X )<f( X > U x><l))^- a ( X ) where x h^ n (x) denotes the outward unit normal to dQ . From this patchy control-Lyapunov function one can construct a robust hybrid feedback stabilizer, at least when the set { u,v.f(x,u) < c } is convex for each real number c and every real vector u , with the following data u,:=k q (x), Cq =(n q \{J ryq n' r )C)0 (12) where k n is defined on C n , continuous and such that (VV q (x),f(x,k q (x)))<-0.5a(x) \/xcC q (n q (x),f(x,k x (x)))<-0.5a(x) V* e (a/2 ? \U r ^ n' r )f]0 The jump set is given by D q =(o\n q )u(\j r ^a r f)o) (14) and the jump map is G q (x) = \) _L " S l ryq ' ' (15) [{reQ:xen' r {]0} xeO\n q With this control, the index increases with each jump except probably the first one. Thus, the number of jumps is finite, and the state converges to the origin, which is also stable. 4.3 Throw-and-catch control In ( Prieur, 2001), it was shown how to combine local and global state feedback to achieve global stabilization and local performance. The idea, which exploits hysteresis switching (Halbaoui et al., 2009b), is completely simple. Two continuous functions, k lobal and k local are shown when the feedback u = k lobal (x) render the origin of the control system x = f(x,u) globally asymptotically stable whereas the feedback u = k local (x) makes the 34 Robust Control, Theory and Applications origin of the control system locally asymptotically stable with basin of attraction containing the open set , which contains the origin. Then we took C local a compact subset of the that contains the origin in its interior and one takes D global to be a compact subset of C local , again containing the origin in its interior and such that, when using the controller k local , trajectories starting in D loM never reach the boundary of C local (Fig. 6). Finally, the hybrid control which achieves global asymptotic stabilization while using the controller k for small signals is as follows w-kJx) C:={ (x,q):xeC\ (16) g{q,x) := toggle (q) D := | (x,q) : x € D q j In the problem of uniting of local and global controllers, one can view the global controller as a type of "bootstrap" controller that is guaranteed to bring the system to a region where another controller can control the system adequately. A prolongation of the idea of combine local and global controllers is to assume the existence of continuous bootstrap controller that is guaranteed to introduce the system, in finite time, in a vicinity of a set of points, not simply a vicinity of the desired final destination (the controller doesn't need to be able to maintain the state in this vicinity); moreover, these sets of points form chains that terminate at the desired final destination and along which controls are known to steer (or "throw") form one point in the chain at the next point in the chain. Moreover, in order to minimize error propagation along a chain, a local stabilizer is known for each point, except perhaps those points at the start of a chain. Those can be employed "to catch" each jet. D ghba! Trajectory due to local controller Fig. 6. Combining local and global controllers 4.4 Supervisory control In this section, we review the supervisory control framework for hybrid systems. One of the main characteristics of this approach is that the plant is approximated by a discrete-event system and the design is carried out in the discrete domain. The hybrid control systems in the supervisory control framework consist of a continuous (state, variable) system to be controlled, also called the plant, and a discrete event controller connected to the plant via an interface in a feedback configuration as shown in (Fig. 7). It is generally assumed that the dynamic behavior of the plant is governed by a set of known nonlinear ordinary differential equations x(t) = f(x(t),r(t)) (17) Robust Control of Hybrid Systems 35 where x e 9? n is the continuous state of the system and re 9? m is the continuous control input. In the model shown in (Fig. 7), the plant contains all continuous components of the hybrid control system, such as any conventional continuous controllers that may have been developed, a clock if time and synchronous operations are to be modeled, and so on. The controller is an event driven, asynchronous discrete event system (DES), described by a finite state automaton. The hybrid control system also contains an interface that provides the means for communication between the continuous plant and the DES controller. Discrete DES Supervisor Envent system IE ▼ [ Interface Control Switch Event recognizer ^ JL_ Continuous varic system ible Controlled system Fig. 7. Hybrid system model in the supervisory control framework. Fig. 8. Partition of the continuous state space. The interface consists of the generator and the actuator as shown in (Fig. 7). The generator has been chosen to be a partitioning of the state space (see Fig. 8). The piecewise continuous command signal issued by the actuator is a staircase signal as shown in (Fig. 9), not unlike the output of a zero-order hold in a digital control system. The interface plays a key role in determining the dynamic behavior of the hybrid control system. Many times the partition of the state space is determined by physical constraints and it is fixed and given. Methodologies for the computation of the partition based on the specifications have also been developed. In such a hybrid control system, the plant taken together with the actuator and generator, behaves like a discrete event system; it accepts symbolic inputs via the actuator and produces symbolic outputs via the generator. This situation is somewhat analogous to the 36 Robust Control, Theory and Applications tjl] t c [2]t c [3] time Fig. 9. Command signal issued by the interface. way a continuous time plant, equipped with a zero-order hold and a sampler, " looks " like a discrete-time plant. The DES which models the plant, actuator, and generator is called the DES plant model. From the DES controller's point of view, it is the DES plant model which is controlled. The DES plant model is an approximation of the actual system and its behavior is an abstraction of the system's behavior. As a result, the future behavior of the actual continuous system cannot be determined uniquely, in general, from knowledge of the DES plant state and input. The approach taken in the supervisory control framework is to incorporate all the possible future behaviors of the continuous plant into the DES plant model. A conservative approximation of the behavior of the continuous plant is constructed and realized by a finite state machine. From a control point of view this means that if undesirable behaviors can be eliminated from the DES plant (through appropriate control policies) then these behaviors will be eliminated from the actual system. On the other hand, just because a control policy permits a given behavior in the DES plant, is no guarantee that that behavior will occur in the actual system. We briefly discuss the issues related to the approximation of the plant by a DES plant model. A dynamical system Z can be described as a triple T;W;B with T cz 9? the time axis, W the signal space, and B cz W T (the set of all functions / : T -» W ) the behavior. The behavior of the DES plant model consists of all the pairs of plant and control symbols that it can generate. The time axis T represents here the occurrences of events. A necessary condition for the DES plant model to be a valid approximation of the continuous plant is that the behavior of the continuous plant model B c is contained in the behavior of the DES plant model, i.e. The main objective of the controller is to restrict the behavior of the DES plant model in order to specify the control specifications. The specifications can be described by a behavior B . Supervisory control of hybrid systems is based on the fact that if undesirable behaviors can be eliminated from the DES plant then these behaviors can likewise be eliminated from the actual system. This is described formally by the relation B rf flB s cB ; spec ' •B c nB s cB : spec (18) and is depicted in (Fig. 10). The challenge is to find a discrete abstraction with behavior Bd which is a approximation of the behavior B c of the continuous plant and for which is possible to design a supervisor in order to guarantee that the behavior of the closed loop system satisfies the specifications B spe c. A more accurate approximation of the plant's behavior can be obtained by considering a finer partitioning of the state space for the extraction of the DES plant. Robust Control of Hybrid Systems 37 Fig. 10. The DES plant model as an approximation. An interesting aspect of the DES plant's behavior is that it is distinctly nondeterministic. This fact is illustrated in (Fig.ll). The figure shows two different trajectories generated by the same control symbol. Both trajectories originate in the same DES plant state p 1 . (Fig.ll) shows that for a given control symbol, there are at least two possible DES plant states that can be reached from p 1 . Transitions within a DES plant will usually be nondeterministic unless the boundaries of the partition sets are invariant manifolds with respect to the vector fields that describe the continuous plant. n Fig. 11. Nondeterminism of the DES plant model. There is an advantage to having a hybrid control system in which the DES plant model is deterministic. It allows the controller to drive the plant state through any desired sequence of regions provided, of course, that the corresponding state transitions exist in the DES plant model. If the DES plant model is not deterministic, this will not always be possible. This is because even if the desired sequence of state transitions exists, the sequence of inputs which achieves it may also permit other sequences of state transitions. Unfortunately, given a continuous-time plant, it may be difficult or even impossible to design an interface that leads to a DES plant model which is deterministic. Fortunately, it is not generally necessary to have a deterministic DES plant model in order to control it. The supervisory control problem for hybrid systems can be formulated and solved when the DES plant model is nondeterministic. This work builds upon the frame work of supervisory control theory used in (Halbaoui et al, 2008) and (Halbaoui et al, 2009a). 5. Robustness to perturbations In control systems, several perturbations can occur and potentially destroy the good behavior for which the controller was designed for. For example, noise in the measurements 38 Robust Control, Theory and Applications of the state taken by controller arises in all implemented systems. It is also common that when a controller is designed, only a simplified model of the system to control exhibiting the most important dynamics is considered. This simplifies the control design in general. However, sensors/ actuators that are dynamics unmodelled can substantially affect the behavior of the system when in the loop. In this section, it is desired that the hybrid controller provides a certain degree of robustness to such disturbances. In the following sections, general statements are made in this regard. 5.1 Robustness via filtered measurements In this section, the case of noise in the measurements of the state of the nonlinear system is considered. Measurement noise in hybrid systems can lead to nonexistence of solutions. This situation can be corrected, at least for the small measurement noise, if under global existence of solutions, C c and D c always " overlap" while ensuring that the stability properties still hold. The "overlap" means that for every £, e O , either £, + e e C c or £, + e e D c all or small e . There exist generally always inflations of C and D that preserve the semiglobal practices asymptotic stability, but they do not guarantee the existence of solutions for small measurement noise. Moreover, the solutions are guaranteed to exist for any locally bounded measurement noise if the measurement noise does not appear in the flow and jump sets. This can be carried out by filtering measures. (Fig. 12) illustrates this scenario. The state x is corrupted by the noise e and the hybrid controller H c measures a filtered version of x + e . Controller ^ I x f Filter 4- — > Hybrid system t; Fig. 12. Closed-loop system with noise and filtered measurements. The filter used for the noisy output y = x + e is considered to be linear and defined by the matrices A* ,Br , and L, , and an additional parameter s * > . It is designed to be asymptotically stable. Its state is denoted by x^ which takes value in R f . At the jumps, x^ is given to the current value of y . Then, the filter has flows given by sir = AcXc + Bey, (19) and jumps given by x}=AJ 1 B f x f + B f y. (20) The output of the filter replaces the state x in the feedback law. The resulting closed-loop system can be interpreted as family of hybrid systems which depends on the parameter 8 r . It is denoted by Hj and is given by Robust Control of Hybrid Systems 39 n cl x = f p (x + K(L f x f ,x c )) X c = fc(^f X f' X c) SrXr = ArXr + B r (X + tf) x + =x X t^ G c( L f X f> X c) xj = -AJ 1 B f (x + e) > (L f x f ,x c )eC c (L f x f ,x c )eD c (21) 5.2 Robustness to sensor and actuator dynamics This section reviews the robustness of the closed-loop H cl when additional dynamics, coming from sensors and actuators, are incorporated. (Fig. 13) shows the closed loop H cl with two additional blocks: a model for the sensor and a model for the actuator. Generally, to simplify the controller design procedure, these dynamics are not included in the model of the system x = f Jx,u) when the hybrid controller H c is conceived. Consequently, it is important to know whether the stability properties of the closed-loop system are preserved, at least semiglobally and practically, when those dynamics are incorporated in the closed loop. The sensor and actuator dynamics are modeled as stable filters. The state of the filter which models the sensor dynamics is given by x s e R n$ with matrices (A S ,B S ,L S ) , the state of the filter that models the actuator dynamics is given by x a e R Ua with matrices (A a ,B a ,L a ) , and £ d > is common to both filters. Augmenting H cl by adding filters and temporal regularization leads to a family H & c f given as follows n cl • X = f P (x>L r *a) x c = fc( L s x s ^c) T = -T + T ^ ■s = 4* s + B s (x + e) £^ 'a ~ ^a x a + B a< L s X s < X c) x + = X x : eG c (L s x s> X c) < = X S < = X a T + =0 ( L s X s> X c)^ C c Or T<T (22) (L s x s ,x c )eD c and x>x where x is a constant satisfying x > x . The following result states that for fast enough sensors and actuators, and small enough temporal regularization parameter, the compact set A is semiglobally practically asymptotically stable. 40 Robust Control, Theory and Applications Controller k~ Actuator t: Sensor Hyi brid system Fig. 13. Closed-loop system with sensor and actuator dynamics. 5.3 Robustness to sensor dynamics and smoothing In many hybrid control applications, the state of the controller is explicitly given as a continuous state £, and a discrete state q eQ:= {l,...,n} , that is, x c := [£, q] T . Where this is the case and the discrete state q chooses a different control law to be applied to the system for for various values of q, then the control law generated by the hybrid controller H c can have jumps when q changes. In many scenarios, it is not possible for the actuator to switch between control laws instantly. In addition, particularly when the control law k(> -,q) is continuous for each q e Q , it is desired to have a smooth transition between them when q changes. k 1 ^ ffi Smoothing Controller <- Sensor Hybrid system ' Fig. 14. Closed-loop system with sensor dynamics and control smoothing. (Fig. 14) shows the closed-loop system, noted that H & c f , resulting from adding a block that makes the smooth transition between control laws indexed by q and indicated by K q . The smoothing control block is modeled as a linear filter for the variable q . It is defined by the parameter s M and the matrices (A U ,B U ,L U ) . The output of the control smoothing block is given by a(x,x c ,L u x u ) x c>i) (23) where for each qeQ,X :R—>[0,1], is continuous and X (q) = l . Note that the output is such that the control laws are smoothly "blended" by the function X . In addition to this block, a filter modeling the sensor dynamics is also incorporated as in section 5.2. The closed loop Hj can be written as Robust Control of Hybrid Systems 41 Kl : x = f p (x + a(x,x c ,L u x u )) X c=fc( L s X s> X c) 4 = T = -T + T 8 M x s = A s x s + B s (x) Vu = A u x u + B uq x + =x < = X s X u = X u T + =0 ( L s x s' x c) gC c or t<t G c (L s x s ,x c ) (24) ( L S * S '*c) eD c and T ^ T 6. Conclusion In this chapter, a dynamic systems approach to analysis and design of hybrid systems has been continued from a robust control point of view. Stability and convergence tools for hybrid systems presented include hybrid versions of the traditional Lyapunov stability theorem and of LaSalle's invar iance principle. The robustness of asymptotic stability for classes of closed-loop systems resulting from hybrid control was presented. Results for perturbations arising from the presence of measurement noise, unmodeled sensor and actuator dynamics, control smoothing. It is very important to have good software tools for the simulation, analysis and design of hybrid systems, which by their nature are complex systems. Researchers have recognized this need and several software packages have been developed. 7. References Rajeev, A.; Thomas, A. & Pei-Hsin, H.(1993). Automatic symbolic verification of embedded systems, In IEEE Real-Time Systems Symposium, 2-11, DOI: 10.1109/REAL.1993.393520 . Dang, T. (2000). Verification et Synthese des Systemes Hybrides. PhD thesis, Institut National Poly technique de Grenoble. Girard, A. (2006). Analyse algorithmique des systemes hybrides. PhD thesis, Universite Joseph Fourier (Grenoble-I). Ancona, F. & Bressan, A. (1999). Patchy vector fields and asymptotic stabilization, ESAIM: Control, Optimisation and Calculus of Variations, 4:445-471, DOI: 10.1051/ cocv:2004003. Byrnes, C. I. & Martin, C. F. (1995). An integral-invariance principle for nonlinear systems, IEEE Transactions on Automatic Control, 983-994, ISSN: 0018-9286. 42 Robust Control, Theory and Applications Cai, C; Teel, A. R. & Goebel, R. (2007). Results on existence of smooth Lyapunov functions for asymptotically stable hybrid systems with nonopen basin of attraction, submitted to the 2007 American Control Conference, 3456 - 3461, ISSN: 0743-1619. Cai, C; Teel, A. R. & Goebel, R. (2006). Smooth Lyapunov functions for hybrid systems Part I: Existence is equivalent to robustness & Part II: (Pre-) asymptotically stable compact sets, 1264-1277, ISSN 0018-9286. Cai, C; Teel, A. R. & Goebel, R. (2005). Converse Lyapunov theorems and robust asymptotic stability for hybrid systems, Proceedings of 24th American Control Conference, 12-17, ISSN: 0743-1619. Chellaboina, V.; Bhat, S. P. & HaddadWH. (2002). An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear Analysis, Chicago, IL, USA, 3116 - 3122,ISBN: 0-7803-5519-9. Goebel, R.; Prieur, C. & Teel, A. R. (2009). smooth patchy control Lyapunov functions. Automatica (Oxford) Y, 675-683 ISSN : 0005-1098. Goebel, R. & Teel, A. R. (2006). Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica, 573-587, DOI: 10.1016/j.automatica.2005.12.019. LaSalle, J. P. (1967). An invariance principle in the theory of stability, in Differential equations and dynamical systems. Academic Press, New York. LaSalle, J. P. (1976) The stability of dynamical systems. Regional Conference Series in Applied Mathematics, SIAM ISBN-13: 978-0-898710-22-9. Lygeros, J.; Johansson, K. H., Simi'c, S. N.; Zhang, J. & Sastry, S. S. (2003). Dynamical properties of hybrid automata. IEEE Transactions on Automatic Control, 2-17 ,ISSN: 0018-9286. Prieur, C. (2001). Uniting local and global controllers with robustness to vanishing noise, Mathematics Control, Signals, and Systems, 143-172, DOI: 10.1007/ PL00009880 Ryan, E. P. (1998). An integral invariance principle for differential inclusions with applications in adaptive control. SIAM Journal on Control and Optimization, 960-980, ISSN 0363- 0129. Sanfelice, R. G.; Goebel, R. & Teel, A. R. (2005). Results on convergence in hybrid systems via detectability and an invariance principle. Proceedings of 2005 American Control Conference, 551-556, ISSN: 0743-1619. Sontag, E. (1989). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 435-443, ISSN: 0018-9286. DeCarlo, R.A.; Branicky, M.S.; Pettersson, S. & Lennartson, B.(2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proc. of IEEE, 1069-1082, ISSN: 0018-9219. Michel, A.N.(1999). Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans. Circuits Syst. - I. Fund. Theory AppL, 120-134,ISSN: 1057-7122. Halbaoui, K.; Boukhetala, D. and Boudjema, F.(2008). New robust model reference adaptive control for induction motor drives using a hybrid con troller. International Symposium on Power Electronics, Electrical Drives, Automation and Motion, Italy, 1109 - 1113 ISBN: 978-1-4244-1663-9. Halbaoui, K.; Boukhetala, D. and Boudjema, F. (2009a). Speed Control of Induction Motor Drives Using a New Robust Hybrid Model Reference Adaptive Controller. Journal of Applied Sciences, 2753-2761, ISSN:18125654. Halbaoui, K.; Boukhetala, D. and Boudjema, F. (2009b). Hybrid adaptive control for speed regulation of an induction motor drive, Archives of Control Sciences, V2. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives Rama K. Yedavalli and Nagini Devarakonda The Ohio State University United States of America 1. Introduction The problem of maintaining the stability of a nominally stable linear time invariant system subject to linear perturbation has been an active topic of research for quite some time. The recent published literature on this v robust stability 7 problem can be viewed mainly from two perspectives, namely i) transfer function (input/ output) viewpoint and ii) state space viewpoint. In the transfer function approach, the analysis and synthesis is essentially carried out in frequency domain, whereas in the state space approach it is basically carried out in time domain. Another perspective that is especially germane to this viewpoint is that the frequency domain treatment involves the extensive use of "polynomial 7 theory while that of time domain involves the use of 'matrix' theory. Recent advances in this field are surveyed in [l]-[2]. Even though in typical control problems, these two theories are intimately related and qualitatively similar, it is also important to keep in mind that there are noteworthy differences between these two approaches ('polynomial' vs 'matrix 7 ) and this chapter (both in parts I and II) highlights the use of the direct matrix approach in the solution to the robust stability and control design problems. 2. Uncertainty characterization and robustness It was shown in [3] that modeling errors can be broadly categorized as i) parameter variations, ii) unmodeled dynamics iii) neglected nonlinearities and finally iv) external disturbances. Characterization of these modeling errors in turn depends on the representation of dynamic system, namely whether it is a frequency domain, transfer function framework or time domain state space framework. In fact, some of these can be better captured in one framework than in another. For example, it can be argued convincingly that real parameter variations are better captured in time domain state space framework than in frequency domain transfer function framework. Similarly, it is intuitively clear that unmodeled dynamics errors can be better captured in the transfer function framework. By similar lines of thought, it can be safely agreed that while neglected nonlinearities can be better captured in state space framework, neglected disturbances can 44 Robust Control, Theory and Applications be captured with equal ease in both frameworks. Thus it is not surprising that most of the robustness studies of uncertain dynamical systems with real parameter variations are being carried out in time domain state space framework and hence in this chapter, we emphasize the aspect of robust stabilization and control of linear dynamical systems with real parameter uncertainty. Stability and performance are two fundamental characteristics of any feedback control system. Accordingly, stability robustness and performance robustness are two desirable (sometimes necessary) features of a robust control system. Since stability robustness is a prerequisite for performance robustness, it is natural to address the issue of stability robustness first and then the issue of performance robustness. Since stability tests are different for time varying systems and time invariant systems, it is important to pay special attention to the nature of perturbations, namely time varying perturbations versus time invariant perturbations, where it is assumed that the nominal system is a linear time invariant system. Typically, stability of linear time varying systems is assessed using Lyapunov stability theory using the concept of quadratic stability whereas that of a linear time invariant system is determined by the Hurwitz stability, i.e. by the negative real part eigenvalue criterion. This distinction about the nature of perturbation profoundly affects the methodologies used for stability robustness analysis. Let us consider the following linear, homogeneous, time invariant asymptotically stable system in state space form subject to a linear perturbation E: x = (A +E)x x(0) = x (1) where Aq is an nxn asymptotically stable matrix and E is the error (or perturbation) matrix. The two aspects of characterization of the perturbation matrix E which have significant influence on the scope and methodology of any proposed analysis and design scheme are i) the temporal nature and ii) the boundedness nature of E. Specifically, we can have the following scenario: i. Temporal Nature: Time invariant error Time varying error E = constant E = E(t) ii. Boundedness Nature: Unstructured Structured vs (Norm bounded) (Elemental bounds) The stability robustness problem for linear time invariant systems in the presence of linear time invariant perturbations (i.e. robust Hurwitz invariance problem) is basically addressed by testing for the negativity of the real parts of the eigenvalues (either in frequency domain or in time domain treatments), whereas the time varying perturbation case is known to be best handled by the time domain Lyapunov stability analysis. The robust Hurwitz invariance problem has been widely discussed in the literature essentially using the polynomial approach [4] -[5]. In this section, we address the time varying perturbation case, mainly motivated by the fact that any methodology which treats the time varying case can always be specialized to the time invariant case but not vice versa. However, we pay a price for the same, namely conservatism associated with the results when applied to the time invariant perturbation case. A methodology specifically tailored to time invariant perturbations is discussed and included by the author in a separate publication [6]. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 45_ It is also appropriate to discuss, at this point, the characterization with regard to the boundedness of the perturbation. In the so called 'unstructured 7 perturbation, it is assumed that one cannot clearly identify the location of the perturbation within the nominal matrix and thus one has simply a bound on the norm of the perturbation matrix. In the 'structured' perturbation, one has information about the location(s) of the perturbation and thus one can think of having bounds on the individual elements of the perturbation matrix. This approach can be labeled as 'Elemental Perturbation Bound Analysis (EPBA)'. Whether 'unstructured' norm bounded perturbation or 'structured' elemental perturbation is appropriate to consider depends very much on the application at hand. However, it can be safely argued that 'structured' real parameter perturbation situation has extensive applications in many engineering disciplines as the elements of the matrices of a linear state space description contain parameters of interest in the evolution of the state variables and it is natural to look for bounds on these real parameters that can maintain the stability of the state space system. 3. Robust stability and control of linear interval parameter systems under state space framework In this section, we first give a brief account of the robust stability analysis techniques in 3.1 and then in subsection 3.2 we discuss the robust control design aspect. 3.1 Robust stability analysis The starting point for the problem at hand is to consider a linear state space system described by x(t) = [A +E]x(t) where x is an n dimensional state vector, asymptotically stable matrix and E is the 'perturbation' matrix. The issue of 'stability robustness measures' involves the determination of bounds on E which guarantee the preservation of stability of (1). Evidently, the characterization of the perturbation matrix E has considerable influence on the derived result. In what follows, we summarize a few of the available results, based on the characterization of E. 1. Time varying (real) unstructured perturbation with spectral norm: Sufficient bound For this case, the perturbation matrix E is allowed to be time varying, i.e. E(t) and a bound on the spectral norm (cr max (£(£)) where o(-) is the singular value of (•)) is derived. When a bound on the norm of E is given, we refer to it as 'unstructured' perturbation. This norm produces a spherical region in parameter space. The following result is available for this case [7]-[8]: <7 max (E(0)<-^— P) ^max^) where P is the solution to the Lyapunov matrix PA +AlP + 2I = (3) See Refs [9], [10], [11] for results related to this case. 46 Robust Control, Theory and Applications 2. Time varying (real) structured variation Case 1: Independent variations (sufficient bound) [12] -[13] E„(^vJE, y (f)| =s V] (4) 1 I J Imax J s = Max i :£ i : 1 € ii < 7 T~ U eii ( 5 ) ] a P U ) ] where P satisfies equation (3) and U y = etj/e. For cases when e,y are not known, one can take Ueij = | A ij \/\A ij | max- (-)m denotes the matrix with all modulus elements and (-) s denotes the symmetric part of (•). 3. Time invariant, (real) structured perturbation E%j = Constant Case i: Independent Variations [13] -[15]: (Sufficient Bounds). For this case, E can be characterized as E = 8^82 (6) where Si and S2 are constant, known matrices and | D t y | < dyd with dy > are given and d > is the unknown. Let LI be the matrix elements Uij = dy. Then the bound on d is given by [13] d<^f- " — - — = »,=»Q H s 2 (y a ,i-A,)" 1 s 1 u) 0>OV L J m J Notice that the characterization of E (with time invariant) in (4) is accommodated by the characterization in [15]. p(-) is the spectral radius of (•). Case ii: Linear Dependent Variation: For this case, E is characterized (as in (6) before), by E = Z- =1 AE, (8) and bounds on 1 13± | are sought. Improved bounds on 1 13± | are presented in [6]. This type of representation represents a 'poly tope of matrices' as discussed in [4]. In this notation, the interval matrix case (i.e. the independent variation case) is a special case of the above representation where Ei contains a single nonzero element, at a different place in the matrix for different z. For the time invariant, real structured perturbation case, there are no computationally tractable necessary and sufficient bounds either for polytope of matrices or for interval matrices (even for a 2 x 2 case). Even though some derivable necessary and sufficient conditions are presented in [16] for any general variation in E (not necessarily linear dependent and independent case), there are no easily computable methods available to determine the necessary and sufficient bounds at this stage of research. So most of the research, at this point of time, seems to aim at getting better (less conservative) sufficient bounds. The following example compares the sufficient bounds given in [13]-[15] for the linear dependent variation case. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 47 Let us consider the example given in [15] in which the perturbed system matrix is given by (A +BKC) Taking the nominally stable matrix to be -2 + h -3 + k 7 -1 + fc, -1 + k 1 -1 + k 2 -4 + k x -2 -1 -3 -1 -1 -4 the error matrix with k\ and fo as the uncertain parameters is given by LL = /C-j -til > /Co-tiO where The following are the bounds on \k\\ and | fo | obtained by [15] and the proposed method. Hv BQ ZK [14] jid[6] "1 1" "0 0" and E 2 = 10 10 1 10 0.815 0.875 1.55 1.75 3.2 Robust control design for linear systems with structured uncertainty Having discussed the robustness analysis issue above, we now switch our attention to the robust control design issue. Towards this direction, we now present a linear robust control design algorithm for linear deterministic uncertain systems whose parameters vary within given bounded sets. The algorithm explicitly incorporates the structure of the uncertainty into the design procedure and utilizes the elemental perturbation bounds developed above. A linear state feedback controller is designed by parameter optimization techniques to maximize (in a given sense) the elemental perturbation bounds for robust stabilization. There is a considerable amount of literature on the aspect of designing linear controllers for linear tine invariant systems with small parameter uncertainty. However, for uncertain systems whose dynamics are described by interval matrices (i.e., matrices whose elements are known to vary within a given bounded interval), linear control design schemes that guarantee stability have been relatively scarce. Reference [17] compares several techniques for designing linear controllers for robust stability for a class of uncertain linear systems. Among the methods considered are the standard linear quadratic regulator (LQR) design, Guaranteed Cost Control (GCC) method of [18], Multistep Guaranteed Cost Control (MGCC) of [17]. In these methods, the weighting on state in a quadratic cost function and the Riccati equation are modified in the search for an appropriate controller. Also, the parameter uncertainty is assumed to enter linearly and restrictive conditions are imposed on the bounding sets. In [18], norm inequalities on the bounding sets are given for stability but 48 Robust Control, Theory and Applications they are conservative since they do not take advantage of the system structure. There is no guarantee that a linear state feedback controller exists. Reference [19] utilizes the concept of 'Matching conditions (MC)' which in essence constrain the manner in which the uncertainty is permitted to enter into the dynamics and show that a linear state feedback control that guarantees stability exists provided the uncertainty satisfies matching conditions. By this method large bounding sets produce large feedback gains but the existence of a linear controller is guaranteed. But no such guarantee can be given for general 'mismatched' uncertain systems. References [20] and [21] present methods which need the testing of definiteness of a Lyapunov matrix obtained as a function of the uncertain parameters. In the multimodel theory approach, [22] considers a discrete set of points in the parameter uncertainty range to establish the stability. This paper addresses the stabilization problem for a continuous range of parameters in the uncertain parameter set (i.e. in the context of interval matrices). The proposed approach attacks the stability of interval matrix problem directly in the matrix domain rather than converting the interval matrix to interval polynomials and then testing the Kharitonov polynomials. Robust control design using perturbation bound analysis [23],[24] Consider a linear, time invariant system described by x = Ax + Bu x(0) = x Where x is nxl state vector, the control u is mxl. The matrix pair (A,B) is assumed to be completely controllable. U=Gx For this case, the nominal closed loop system matrix is given by A=A+BG , G and d-1 KA + A T K - KB-Q-B T K + Q = Pc and A is asymptotically stable. Here G is the Riccati based control gain where Q,and Ro are any given weighting matrices which are symmetric, positive definite and p c is the design variable. The main interest in determining G is to keep the nominal closed loop system stable. The reason Riccati approach is used to determine G is that it readily renders (A+BG) asymptotically stable with the above assumption on Q and Ro. Now consider the perturbed system with linear time varying perturbations Ea(0 and Eg(f) respectively in matrices A and B i.e., x = [A + E A (t)]x(t) + [B + E B (t)]u(t) Let AA and AB be the perturbation matrices formed by the maximum modulus deviations expected in the individual elements of matrices A and B respectively. Then one can write (Absolute variation) Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 49_ where e a is the maximum of all the elements in AA and z\, is the maximum of all elements in AB. Then the total perturbation in the linear closed loop system matrix of (10) with nominal control u = Gx is given by A = AA + ABG m =s a U ea +s b U eh G m Assuming the ratio is s h js a = s known, we can extend the main result of equation (3) to the linear state feedback control system of (9) and (10) and obtain the following design observation. Design observation 1: The perturbed linear system is stable for all perturbations bounded by s a and s h if ■ = /* (9) and s h <s ju where °^[ P m{ U ea + £U eb G m)\ P(A + BG) + (A + BG) T P + 2I n = Remark: If we suppose AA = 0, AB = and expect some control gain perturbations AG, where we can write then stability is assured if AG = sAle R (10) s g < t^ - = M* (!1) °"max \* mPnr*- eg ) In this context jug can be regarded as a "gain margin". For a given s ai j and %• , one method of designing the linear controller would be to determine G of (3.10) by varying p c of (3.10) such that u is maximum. For an aircraft control example which utilizes this method, see Reference [9]. 4. Robust stability and control of linear interval parameter systems using ecological perspective It is well recognized that natural systems such as ecological and biological systems are highly robust under various perturbations. On the other hand, engineered systems can be made highly optimal for good performance but they tend to be non-robust under perturbations. Thus, it is natural and essential for engineers to delve into the question of as to what the underlying features of natural systems are, which make them so robust and then try to apply these principles to make the engineered systems more robust. Towards this objective, the interesting aspect of qualitative stability in ecological systems is considered in particular. The fields of population biology and ecology deal with the analysis of growth and decline of populations in nature and the struggle of species to predominate over one another. The existence or extinction of a species, apart from its own effect, depends on its interactions with various other species in the ecosystem it belongs to. Hence the type of interaction is very critical to the sustenance of species. In the following sections these 50 Robust Control, Theory and Applications interactions and their nature are thoroughly investigated and the effect of these qualitative interactions on the quantitative properties of matrices, specifically on three matrix properties, namely, eigenvalue distribution, normality/ condition number and robust stability are presented. This type of study is important for researchers in both fields since qualitative properties do have significant impact on the quantitative aspects. In the following sections, this interrelationship is established in a sound mathematical framework. In addition, these properties are exploited in the design of controllers for engineering systems to make them more robust to uncertainties such as described in the previous sections. 4.1 Robust stability analysis using principles of ecology 4.1.1 Brief review of ecological principles In this section a few ecological system principles that are of relevance to this chapter are briefly reviewed. Thorough understanding of these principles is essential to appreciate their influence on various mathematical results presented in the rest of the chapter. In a complex community composed of many species, numerous interactions take place. These interactions in ecosystems can be broadly classified as i) Mutualism, ii) Competition, iii) Commensalism/Ammensalism and iv) Predation (Parasitism). Mutualism occurs when both species benefit from the interaction. When one species benefits/ suffers and the other one remains unaffected, the interaction is classified as Commensalism/Ammensalism. When species compete with each other, that interaction is known as Competition. Finally, if one species is benefited and the other suffers, the interaction is known as Predation (Parasitism). In ecology, the magnitudes of the mutual effects of species on each other are seldom precisely known, but one can establish with certainty, the types of interactions that are present. Many mathematical population models were proposed over the last few decades to study the dynamics of eco/bio systems, which are discussed in textbooks [25]- [26]. The most significant contributions in this area come from the works of Lotka and Volterra. The following is a model of a predator-prey interaction where x is the prey and y is the predator. x = xf(x,y) jk »> (12) y = yg( x >y) where it is assumed that df(x,y) / dy < and dg(x,y) / dx > This means that the effect of y on the rate of change of x ( x ) is negative while the effect of x on the rate of change of y ( y ) is positive. The stability of the equilibrium solutions of these models has been a subject of intense study in life sciences [27]. These models and the stability of such systems give deep insight into the balance in nature. If a state of equilibrium can be determined for an ecosystem, it becomes inevitable to study the effect of perturbation of any kind in the population of the species on the equilibrium. These small perturbations from equilibrium can be modeled as linear state space systems where the state space plant matrix is the / J ac °Dian / . This means that technically in the Jacobian matrix, one does not know the actual magnitudes of the partial derivatives but their signs are known with certainty. That is, the nature of the interaction is known but not the strengths of those interactions. As mentioned previously, there are four classes of interactions and after linearization they can be represented in the following Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 51 Interaction type Digraph representation Matrix representation Mutualism O) '* +~ + * Competition GC© _ * Commensalism ay-^s '* + ~ * Ammensalism &^S * Predation (Parasitism) (3Z© '* +~ _ * Table 1. Types of interactions between two species in an ecosystem In Table 1, column 2 is a visual representation of such interactions and is known as a directed graph or 'digraph' [28] while column 3 is the matrix representation of the interaction between two species. '*' represents the effect of a species on itself. In other words, in the Jacobian matrix, the 'qualitative' information about the species is represented by the signs +, - or 0. Thus, the (i,j) th entry of the state space (Jacobian) matrix simply consists of signs +, -, or 0, with the + sign indicating species j having a positive influence on species i, - sign indicating negative influence and indicating no influence. The diagonal elements give information regarding the effect of a species on itself. Negative sign means the species is 'self -regulatory', positive means it aids the growth of its own population and zero means that it has no effect on itself. For example, in the Figure 1 below, sign pattern matrices Ai and Ai are the Jacobian form while D\ and D2 are their corresponding digraphs. b) A 2 a) 0-000 + 0-00 + - - + 0- + + + -00 - - Fig. 1. Various sign patterns and their corresponding digraphs representing ecological systems; a) three species system b) five species system 52 Robust Control, Theory and Applications 4.1.2 Qualitative or sign stability Since traditional mathematical tests for stability fail to analyze the stability of such ecological models, an extremely important question then, is whether it can be concluded, just from this sign pattern, whether the system is stable or not. If so, the system is said to be 'qualitatively stable' [29-31]. In some literature, this concept is also labeled as 'sign stability'. In what follows, these two terms are used interchangeably. It is important to keep in mind that the systems (matrices) that are qualitatively (sign stable) stable are also stable in the ordinary sense. That is, qualitative stability implies Hurwitz stability (eigenvalues with negative real part) in the ordinary sense of engineering sciences. In other words, once a particular sign matrix is shown to be qualitatively (sign) stable, any magnitude can be inserted in those entries and for all those magnitudes the matrix is automatically Hurwitz stable. This is the most attractive feature of a sign stable matrix. However, the converse is not true. Systems that are not qualitatively stable can still be stable in the ordinary sense for certain appropriate magnitudes in the entries. From now on, to distinguish from the concept of 'qualitative stability' of life sciences literature, the label of 'quantitative stability' for the standard Hurwitz stability in engineering sciences is used. These conditions in matrix theory notation are given below i. a u < V i ii. and a u < for at least one i iii. flfflfl <0 Vz,; i * j iv. a f/ -a- fc %...a mi = for any sequence of three or more distinct indices i,j,k, . . .m. v. Det(A) * vi. Color test (Elaborated in [32],[33]) Note: In graph theory a • -a •■ are referred to as /-cycles and a i :a- ]i a ]d ...a mi are referred to as ^-cycles. In [34], [35], /-cycles are termed 'interactions' while fc-cycles are termed 'interconnections' (which essentially are all zero in the case of sign stable matrices). With this algorithm, all matrices that are sign stable can be stored apriori as discussed in [36]. If a sign pattern in a given matrix satisfies the conditions given in the above papers (thus in the algorithm), it is an ecological stable sign pattern and hence that matrix is Hurwitz stable for any magnitudes in its entries. A subtle distinction between 'sign stable' matrices and 'ecological sign stable' matrices is now made, emphasizing the role of nature of interactions. Though the property of Hurwitz stability is held in both cases, ecosystems sustain solely because of interactions between various species. In matrix notation this means that the nature of off-diagonal elements is essential for an ecosystem. Consider a strictly upper triangular 3x3 matrix A = - + - - Cj5^- From quantitative viewpoint, it is seen that the matrix is Hurwitz stable for any magnitudes in the entries of the matrix. This means that it is indeed (qualitatively) sign stable. But since there is no predator-prey link and in fact no link at all between species 1&2 and 3&2, such a Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 53_ digraph cannot represent an ecosystem. Therefore, though a matrix is sign stable, it need not belong to the class of ecological sign stable matrices. In Figure 2 below, these various classes of sign patterns and the corresponding relationship between these classes is depicted. So, every ecological sign stable sign pattern is sign stable but the converse is not true. With this brief review of ecological system principles, the implications of these ecological qualitative principles on three quantitative matrix theory properties, namely eigenvalues, normality/ condition number and robust stability are investigated. In particular, in the next section, new results that clearly establish these implications are presented. As mentioned in the previous section, the motivation for this study and analysis is to exploit some of these desirable features of ecological system principles to design controllers for engineering systems to make them more robust. All sign patterns All stable sign patterns All ecologically stable sign patterns Fig. 2. Classification of sign patterns 4.2 Ecological sign stability and its implications in quantitative matrix theory In this major section of this chapter, focusing on the ecological sign stability aspect discussed above, its implications in the quantitative matrix theory are established. In particular, the section offers three explicit contributions to expand the current knowledge base, namely i) Eigenvalue distribution of ecological sign stable matrices ii) Normality/ Condition number properties of sign stable matrices and iii) Robustness properties of sign stable matrices. These three contributions in turn help in determining the role of magnitudes in quantitative ecological sign stable matrices. This type of information is clearly helpful in designing robust controllers as shown in later sections. With this motivation, a 3-species ecosystem is thoroughly analyzed and the ecological principles in terms of matrix properties that are of interest in engineering systems are interpreted. This section is organized as follows: First, new results on the eigenvalue distribution of ecological sign stable matrices are presented. Then considering ecological systems with only predation-prey type interactions, it is shown how selection of appropriate magnitudes in these interactions imparts the property of normality (and thus highly desirable condition numbers) in matrices. In what follows, for each of these cases, concepts are first discussed from an ecological perspective and then later the resulting matrix theory implications from a quantitative perspective are presented Stability and eigenvalue distribution Stability is the most fundamental property of interest to all dynamic systems. Clearly, in time invariant matrix theory, stability of matrices is governed by the negative real part 54 Robust Control, Theory and Applications nature of its eigenvalues. It is always useful to get bounds on the eigenvalue distribution of a matrix with as little computation as possible, hopefully as directly as possible from the elements of that matrix. It turns out that sign stable matrices have interesting eigenvalue distribution bounds. A few new results are now presented in this aspect. In what follows, the quantitative matrix theory properties for an n-species ecological system is established, i.e., an nxn sign stable matrix with predator-prey and commensal/ ammensal interactions is considered and its eigenvalue distribution is analyzed. In particular, various cases of diagonal elements' nature, which are shown to possess some interesting eigenvalue distribution properties, are considered. Bounds on real part of eigenvalues Based on several observations the following theorem for eigenvalue distribution along the real axis is stated. Theorem 1 [37] (Case of all negative diagonal elements): For all nxn sign stable matrices, with all negative diagonal elements, the bounds on the real farts of the eigenvalues are given as follows: The lower bound on the magnitude of the real part is given by the minimum magnitude diagonal element and the upper bound is given by the maximum magnitude diagonal element in the matrix. That is, for an nxn ecological sign stable matrix A = [a^], \a u \ . <\Re(X)\ . <\Re(X)\ <\a u \ (13) I "limn I v /Imin I v 'Imax I " Imax v ' Corollary (Case of some diagonal elements being zero): If the ecological sign stable matrix has zeros on the diagonal, the bounds are given by \a ti \ . (=0)<|Re(/l)| . <\Re(X)\ <\a u \ (14) I "ImmV / I V /Imin I v 'Imax I "Imax v ' The sign pattern in Example 1 has all negative diagonal elements. In this example, the case discussed in the corollary where one of the diagonal elements is zero, is considered. This sign pattern is as shown in the matrix below. A = 0-0 + + Bounds on imaginary part of eigenvalues [38] Similarly, the following theorem can be stated for bounds on the imaginary parts of the eigenvalues of an nxn matrix. Before stating the theorem, we present the following lemma. Theorem 2 For all nxn ecologically sign stable matrices, bound on the imaginary part of the eigenvalues is given by ju\ . = llmag (\)\ = V -a::a H Mi*i (15) ^hmagss I & V i/| ma x \ \ *-* l J J 1 J Above results are illustrated in figure 3. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 55 6 - -4.5 -3.5 -2.5 -1.5 -0.5 0.5 Fig. 3. Eigenvalue distribution for sign stable matrices Theorem 3 For all nxn matrices, with all k-cycles being zero and with only commensal or ammensal interactions, the eigenvalues are simply the diagonal elements. It is clear that these theorems offer significant insight into the eigenvalue distribution of nxn ecological sign stable matrices. Note that the bounds can be simply read off from the magnitudes of the elements of the matrices. This is quite in contrast to the general quantitative Hurwitz stable matrices where the lower and upper bounds on the eigenvalues of a matrix are given in terms of the singular values of the matrix and/ or the eigenvalues of the symmetric part and skew-symmetric parts of the matrices (using the concept of field of values), which obviously require much computation, and are complicated functions of the elements of the matrices. Now label the ecological sign stable matrices with magnitudes inserted in the elements as 'quantitative ecological sign stable matrices'. Note that these magnitudes can be arbitrary in each non zero entry of the matrix! It is interesting and important to realize that these bounds, based solely on sign stability, do not reflect diagonal dominance, which is the typical case with general Hurwitz stable matrices. Taking theorems 4, 5, 6 and their respective corollaries into consideration, we can say that it is the 'diagonal connectance' that is important in these quantitative ecological sign stable matrices and not the 'diagonal dominance' which is typical in the case of general Hurwitz stable matrices. This means that interactions are critical to system stability even in the case of general nxn matrices. Now the effect on the quantitative property of normality is presented. Normality and condition number Based on this new insight on the eigenvalue distribution of sign stable matrices, other matrix theory properties of sign stable matrices are investigated. The first quantitative matrix theory property is that of normality / condition number. But this time, the focus is only on ecological sign stable matrices with pure predator-prey links with no other types of interactions. 56 Robust Control, Theory and Applications A zero diagonal element implies that a species has no control over its growth/ decay rate. So in order to regulate the population of such a species, it is essential that, in a sign stable ecosystem model, this species be connected to at least one predator-prey link. In the case where all diagonal elements are negative, the matrix represents an ecosystem with all self- regulating species. If every species has control over its regulation, a limiting case for stability is a system with no inter speciel interactions. This means that there need not be any predator-prey interactions. This is a trivial ecosystem and such matrices actually belong to the only 'sign-stable' set, not to ecological sign stable set. Apart from the self-regulatory characteristics of species, the phenomena that contribute to the stability of a system are the type of interactions. Since a predator-prey interaction has a regulating effect on both the species, predator-prey interactions are of interest in this stability analysis. In order to study the role played by these interactions, henceforth focus is on systems with n-1 pure predator-prey links in specific places. This number of links and the specific location of the links are critical as they connect all species at the same time preserving the property of ecological sign stability. For a matrix A, pure predator-prey link structure implies that 1. 4 7 A 7 ,<0 Mi,] 2. A^ = iff A, =A ;7 =0 Hence, in what follows, matrices with all negative diagonal elements and with pure predator-prey links are considered. Consider sign stable matrices with identical diagonal elements (negative) and pure predator-prey links of equal strengths. Normality in turn implies that the modal matrix of the matrix is orthogonal resulting in it having a condition number of one, which is an extremely desirable property for all matrices occurring in engineering applications. The property of normality is observed in higher order systems too. An ecologically sign stable matrix with purely predator-prey link interactions is represented by the following digraph for a 5-species system. The sign pattern matrix A represents this digraph. + A = - + 00 0" - - + - - + 0-- + 0- - Theorem 4 An nxn matrix A with equal diagonal elements and equal predation prey interaction strengths for each predation-prey link is a normal matrix. The property of k=1 is of great significance in the study of robustness of stable matrices. This significance will be explained in the next section eventually leading to a robust control design algorithm Robustness The third contribution of this section is related to the connection between ecological sign stability and robust stability in engineering systems. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 57_ As mentioned earlier, the most interesting feature of ecological sign stable matrices is that the stability property is independent of the magnitude information in the entries of the matrix. Thus the nature of interactions, which in turn decide the signs of the matrix entries and their locations in the matrix, are sufficient to establish the stability of the given sign matrix. Clearly, it is this independence (or non-dependence) from magnitude information that imparts the property of robust stability to engineering systems. This aspect of robust stability in engineering systems is elaborated next from quantitative matrix theory point of view. Robustness as a result of independence from magnitude information In mathematical sciences, the aspect of 'robust stability' of families of matrices has been an active topic of research for many decades. This aspect essentially arises in many applications of system and control theory. When the system is described by linear state space representation, the plant matrix elements typically depend on some uncertain parameters which vary within a given bounded interval. Robust stability analysis of a class of interval matrices [39]: Consider the 'interval matrix family' in which each individual element varies independently within a given interval. Thus the interval matrix family is denoted by Ae[A L ,A li ] as the set of all matrices A that satisfy (a l ) <A f; <(A u ) for every i,j >l] J v >l] Now, consider a special 'class of interval matrix family' in which for each element that is varying, the lower bound i.e. (A L )y and the wpiper bound i.e. (A u )ij are of the same sign. For example, consider the interval matrix given by 2 < a 12 < 5 a 12 a 13 l<a 13 <4 -3 < a 21 < -1 %3_ -4 < a 31 < -2 -5 < %, < -0.5 with the elements an, an, 021, 031 and #33 being uncertain varying in some given intervals as follows: Qualitative stability as a 'sufficient condition' for robust stability of a class of interval matrices: A link between life sciences and engineering sciences It is clear that ecological sign stable matrices have the interesting feature that once the sign pattern is a sign stable pattern, the stability of the matrix is independent of the magnitudes of the elements of the matrix. That this property has direct link to stability robustness of matrices with structured uncertainty was recognized in earlier papers on this topic [32] and [33]. In these papers, a viewpoint was put forth that advocates using the 'qualitative stability' concept as a means of achieving 'robust stability' in the standard uncertain matrix theory and offer it as a 'sufficient condition' for checking the robust stability of a class of interval matrices. This argument is illustrated with the following examples. Consider the above given 'interval matrix'. Once it is recognized that the signs of the interval entries in the matrix are not changing (within the given intervals), the sign matrix can be formed. The v sign' matrix for this interval matrix is given by "0 + + - - - 58 Robust Control, Theory and Applications A-- The above 'sign 7 matrix is known to be 'qualitative (sign) stable'. Since sign stability is independent of magnitudes of the entries of the matrix, it can be concluded that the above interval matrix is robustly stable in the given interval ranges. Incidentally, if the 'vertex algorithm 7 of [40] is applied for this problem, it can be also concluded that this 'interval matrix family' is indeed Hurwitz stable in the given interval ranges. In fact, more can be said about the 'robust stability' of this matrix family using the 'sign stability' application. This matrix family is indeed robustly stable, not only for those given interval ranges above, but it is also robustly stable for any large 'interval ranges' in those elements as long as those interval ranges are such that the elements do not change signs in those interval ranges. In the above discussion, the emphasis was on exploiting the sign pattern of a matrix in robust stability analysis of matrices. Thus, the tolerable perturbations are direction sensitive. Also, no perturbation is allowed in the structural zeroes of the ecological sign stable matrices. In what follows, it is shown that ecological sign stable matrices can still possess superior robustness properties even under norm bounded perturbations, in which perturbations in structural zeroes are also allowed in ecological sign stable matrices. Towards this objective, the stability robustness measures of linear state space systems as discussed in [39] and [2] are considered. In other words, a linear state space plant matrix A, which is assumed to be Hurwitz stable, is considered. Then assuming a perturbation matrix E in the A matrix, the question as to how much of norm of the perturbation matrix E can be tolerated to maintain stability is asked. Note that in this norm bounded perturbation discussion, the elements of the perturbation matrix can vary in various directions without any restrictions on the signs of the elements of that matrix. When bounds on the norm of E are given to maintain stability, it is labeled as robust stability for unstructured, norm bounded uncertainty. We now briefly recall two measures of robustness available in the literature [2] for robust stability of time varying real parameter perturbations. Norm bounded robustness measures Consider a given Hurwitz stable matrix Aq with perturbation E such that A = A + E (16) where A is any one of the perturbed matrices. A sufficient bound u for the stability of the perturbed system is given on the spectral norm of the perturbation matrix as ^ >(A(A )) L ^ (i7) K K where a s is the real part of the dominant eigenvalue, also known as stability degree and k is the condition number of the modal matrix of Aq. Theorem 5[38] |ReWA NN ))| min >|Re(A(B NN ))| min ^ i.e.,fi(A NN )>fi(B NN ) Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 59_ In other words, a unit norm, normal ecological sign stable matrix is more robust that a unit norm, normal non-ecological sign stable Hurwitz stable matrix. The second norm bound based on the solution of the Lyapunov matrix equation [7] is given as °max( P ) where P is the solution of the Lyapunov equation of the nominal stable matrix Ao given by A%P + PA +2I = Based on this bound, the following Lemma is proposed: Theorem 6 The norm bound ju on a target SS matrix S is d, where d is the magnitude of diagonal element of S i.e., = d (20) This means that for any given value of \i v , we can, by mere observation, determine a corresponding stable matrix A! This gives impetus to design controllers that drive the closed loop system to a target matrix. Towards this objective, an algorithm for the design of a controller based on concepts from ecological sign stability is now presented. 4.3 Robust control design based on ecological sign stability Extensive research in the field of robust control design has lead to popular control design methods in frequency domain such as H^ and //-synthesis., Though these methods perform well in frequency domain, they become very conservative when applied to the problem of accommodating real parameter uncertainty. On the other hand, there are very limited robust control design methods in time domain methods that explicitly address real parameter uncertainty [41-47]. Even these very few methods tend to be complex and demand some specific structure to the real parameter uncertainty (such as matching conditions). Therefore, as an alternative to existing methods, the distinct feature of this control design method inspired by ecological principles is its problem formulation in which the robustness measure appears explicitly in the design methodology. 4.3.1 Problem formulation The problem formulation for this novel control design method is as follows: For a given linear system x(t) = Ax(t) + Bu(t) (21) design a full-state feedback controller u = Gx (22) 60 Robust Control, Theory and Applications where the closed loop system - B C = A nxm mxn clnx (23) possesses a desired robustness bound ]i (there is no restriction on the value this bound can assume). Since eigenvalue distribution, condition number (normality) and robust stability properties have established the superiority of target matrices, they become an obvious choice for the closed loop system matrix A c i . Note that the desired bound \i= \i& = \i v . Therefore, the robust control design method proposed in the next section addresses the three viewpoints of robust stability simultaneously^ 4.3.2 Robust control design algorithm Consider the LTI system x = Ax + Bu Then, for a full-state feedback controller, the closed loop system matrix is given by A, nxm mxn clnxn\ t) Let A rl -A = A„ (24) The control design method is classified as follows: 1. Determination of Existence of the Controller [38] 2. Determination of Appropriate Closed loop System [38] 3. Determination of Control Gain Matrix[48] Following example illustrates this simple and straightforward control design method. Application: Satellite formation flying control problem The above control algorithm is now illustrated for the application discussed in [32], [33] and [49]. X X = y _y_ 3co 2 where x,x,y and y are the state variables, T x and T y are the control variables. For example, when a = 1, the system becomes 1 0" X "0 0" 1 2co X y + 1 ± x T V 2co _y_ 1 (25) 1 0" "0 0" 1 2 and B = 1 3 -2 1 Clearly, the first two rows of A c \ cannot be altered and hence a target matrix with all non- zero elements cannot be achieved. Therefore, a controller such that the closed loop system has as many features of a target SS matrix as possible is designed as given below. Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 61 Accordingly, an ecological sign stable closed loop system is chosen such that i The closed loop matrix has as many pure predator-prey links as possible. ii It also has as many negative diagonal elements as possible. Taking the above points into consideration, the following sign pattern is chosen which is appropriate for the given A and B matrices: + 0" + - - + Ai = - - - 1 -1 -1 -1 -2 The magnitudes of the entries of the above sign matrix are decided by the stability robustness analysis theorem discussed previously i.e., i All non-zero an are identical. ii aij = - Oji for all non-zero a^ else a^ = a^ = Hence, all the pure predator-prey links are of equal interaction strengths and the non-zero diagonal elements have identical self -regulatory intensities. Using the algorithm given above, the gain matrix is computed as shown below. From the algorithm, -1.0 -4.0 -1.0 -1.0 The closed loop matrix A c i (= A+BG es ) is sign-stable and hence can tolerate any amount of variation in the magnitudes of the elements with the sign pattern kept constant. In this application, it is clear that all non-zero elements in the open loop matrix (excluding elements A13 and A24 since they are dummy states used to transform the system into a set of first order differential equations) are functions of the angular velocity co. Hence, real life perturbations in this system occur only due to variation in angular velocity co. Therefore, a perturbed satellite system is simply an A matrix generated by a different co. This means that not every randomly chosen matrix represents a physically perturbed system and that for practical purposes, stability of the matrices generated as mentioned above (by varying co) is sufficient to establish the robustness of the closed loop system. It is only because of the ecological perspective that these structural features of the system are brought to light. Also, it is the application of these ecological principles that makes the control design for satellite formation flying this simple and insightful. Ideally, we would like A t to be the eventual closed loop system matrix. However, it may be difficult to achieve this objective for any given controllable pair (A,B). Therefore, we propose to achieve a closed loop system matrix that is close to A t . Thus the closed loop system is expressed as A d =A + BG = A t + AA (26) Noting that ideally we like to aim for AA = 0, we impose this condition. Then, A c \ = A t A+BG. i. When B is square and invertible: As given previously, A cl =A t and G = B~ 1 (A-A t ) 62 Robust Control, Theory and Applications ii. When B is not square, but has full rank: Consider Bt, the pseudo inverse of B where, for B nxm , if n > m, Bt = {b t bJ B T Then G= Bt(A-A f ) Because of errors associated with pseudo inverse operation, the expression for the closed loop system is as follows [34]: A t +AE = A + BG A t +AE = A + B^bJ 1 B T (A t - A) (27) Let B{B T B) 1 B T =B aug ThenzlE = (A-A t ) + B^(A t -A) = -(A t -A) + B aMg (A t -A) = (B^-j)(A t -A) :.AE = (B mg -l)(A-A t ) (28) which should be as small as possible. Therefore, the aim is to minimize the norm of AE. Thus, for a given controllable pair {A,B), we use the elements of the desired closed loop matrix A t as design variables to minimize the norm of AE. We now apply this control design method to aircraft longitudinal dynamics problem. Application: Aircraft flight control Consider the following short period mode of the longitudinal dynamics of an aircraft [50]. -0.334 -2.52 1 -0.387 -0.027 -2.6 (29) Open loop A Target matrix At Close loop A c i Matrix "-0.334 1 -2.52 -0.387 " -0.3181 1.00073" -1.00073 -0.3181 " -0.3182 1.00073" -1.00073 -0.319 Eigenvalues -0.3605 + 7 1.5872 -0.3605 -;1.5872 "-0.3181 + /1.00073" -0.3181-/1.00073 -0.31816 + 7I. 000722 -0.31816 -7I. 000722 Norm bound 0.2079 0.3181 0.3181426 The open loop matrix properties are as follows: Note that the open loop system matrix is stable and has a Lyapunov based robustness bound flop = 0.2079. Now for the above controllable pair {A,B), we proceed with the proposed control design procedure discussed before, with the target PS matrix A t elements as design variables, which very quickly yields the following results: A t is calculated by minimizing the norm of a max (AE). Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives 63 Here cr max (zlE) = 1.2381 xlCT 4 For this value, following are the properties of the target matrix. From the expression for G, we get G = [-0.5843 -0.0265] With this controller, the closed loop matrix A c i is determined. It is easy to observe that the eventual closed loop system matrix is extremely close to the target PS matrix (since a max (AE) «0) and hence the resulting robustness bounds can be simply read off from the diagonal elements of the target SS matrix, which in this example is also equal to the eventual closed loop system matrix. As expected, this robustness measure of the closed loop system is appreciably greater than the robustness measure of the open loop system. This robust controller methodology thus promises to be a desirable alternative to the other robustness based controllers encompassing many fields of application. 5. Conclusions and future directions In this book chapter, robust control theory is presented essentially from a state space perspective. We presented the material in two distinct parts. In the first part of the chapter, robust control theory is presented from a quantitative (engineering) perspective, making extensive use of state space models of dynamic systems. Both robust stability analysis as well as control design were addressed and elaborated. Robust stability analysis involved studying and quantifying the tolerable bounds for maintaining the stability of a nominally stable dynamic system. Robust control design dealt with the issue of synthesizing a controller to keep the closed loop systems stable under the presence of a given set of perturbations. This chapter focused on characterizing the perturbations essentially as v real parameter 7 perturbations and all the techniques presented accommodate this particular modeling error. In the second part of the chapter, robustness is treated from a completely new perspective, namely from concepts of Population (Community) Ecology, thereby emphasizing the v qualitative' nature of the stability robustness problem. In this connection, the analysis and design aspects were directed towards studying the role of Signs' of the elements of the state space matrices in maintaining the stability of the dynamic system. Thus the concept of 'sign stability 7 from the field of ecology was brought out to the engineering community. This concept is relatively new to the engineering community. The analysis and control design for engineering systems using ecological principles as presented in this chapter is deemed to spur exciting new research in this area and provide new directions for future research. In particular, the role of v interactions and interconnections' in engineering dynamic systems is shown to be of paramount importance in imparting robustness to the system and more research is clearly needed to take full advantage of these promising ideas. This research is deemed to pave the way for fruitful collaboration between population (community) ecologists and control systems engineers. 6. References [1] Dorato, P., "Robust Control", IEEE Press, New York, N.Y., 1987 [2] Dorato, P., and Yedavalli, R. K., (Eds) Recent Advances in Robust Control, IEEE Press, 1991, pp. 109-111. 64 Robust Control, Theory and Applications [3] Skelton, R. "Dynamic Systems Control, " John Wiley and Sons, New York, 1988 [4] Barmish, B. R., "New Tools for Robustness of Linear Systems", Macmillan Publishing Company, New York, 1994 [5] Bhattacharya, S. P., Chapellat, H., and Keel, L. H., "Robust Control: The Parameteric Approach", Prentice Hall, 1995 [6] Yedavalli, R. 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A., "Satellite Formation Flying and Control Design Based on Hybrid Control System Stability Analysis," Proceedings of the American Control Conference, June 2000, pp. 2210. Nelson, R., Flight Stability and Automatic Control. McGraw Hill. Chap. 1998. Part 2 H-infinity Control Robust Hoc PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis Endra Joelianto Bandung Institute of Technology Indonesia 1. Introduction PID controller has been extensively used in industries since 1940s and still the most often implemented controller today. The PID controller can be found in many application areas: petroleum processing, steam generation, polymer processing, chemical industries, robotics, unmanned aerial vehicles (UAVs) and many more. The algorithm of PID controller is a simple, single equation relating proportional, integral and derivative parameters. Nonetheless, these provide good control performance for many different processes. This flexibility is achieved through three adjustable parameters of which values can be selected to modify the behaviour of the closed loop system. A convenient feature of the PID controller is its compatibility with enhancement that provides higher capabilities with the same basic algorithm. Therefore the performance of a basic PID controller can be improved through judicious selection of these three values. Many tuning methods are available in the literature, among with the most popular method the Ziegler-Nichols (Z-N) method proposed in 1942 (Ziegler & Nichols, 1942). A drawback of many of those tuning rules is that such rules do not consider load disturbance, model uncertainty, measurement noise, and set-point response simultaneously. In general, a tuning for high performance control is always accompanied by low robustness (Shinskey, 1996). Difficulties arise when the plant dynamics are complex and poorly modeled or, specifications are particularly stringent. Even if a solution is eventually found, the process is likely to be expensive in terms of design time. Varieties of new methods have been proposed to improve the PID controller design, such as analytical tuning (Boyd & Barrat, 1991; Hwang & Chang, 1987), optimization based (Wong & Seborg, 1988; Boyd & Barrat, 1991; Astrom & Hagglund, 1995), gain and phase margin (Astrom & Hagglund, 1995; Fung et al., 1998). Further improvement of the PID controller is sought by applying advanced control designs (Ge et al., 2002; Hara et al., 2006; Wang et al., 2007; Goncalves et al, 2008). In order to design with robust control theory, the PID controller is expressed as a state feedback control law problem that can then be solved by using any full state feedback robust control synthesis, such as Guaranteed Cost Design using Quadratic Bound (Petersen et al, 2000), Hoo synthesis (Green & Limebeer, 1995; Zhou & Doyle, 1998), Quadratic Dissipative Linear Systems (Yuliar et al., 1997) and so forth. The PID parameters selection by 70 Robust Control, Theory and Applications transforming into state feedback using linear quadratic method was first proposed by Williamson and Moore in (Williamson & Moore, 1971). Preliminary applications were investigated in (Joelianto & Tomy, 2003) followed the work in (Joelianto et al., 2008) by extending the method in (Williamson & Moore, 1971) to Hoc synthesis with dissipative integral backstepping. Results showed that the robust Hoc PID controllers produce good tracking responses without overshoot, good load disturbance responses, and minimize the effect of plant uncertainties caused by non-linearity of the controlled systems. Although any robust control designs can be implemented, in this paper, the investigation is focused on the theory of parameter selection of the PID controller based on the solution of robust Hoo which is extended with full state dissipative control synthesis and integral backstepping method using an algebraic Riccati inequality (ARI). This paper also provides detailed derivations and improved conditions stated in the previous paper (Joelianto & Tomy, 2003) and (Joelianto et al., 2008). In this case, the selection is made via control system optimization in robust control design by using linear matrix inequality (LMI) (Boyd et al., 1994; Gahinet & Apkarian, 1994). LMI is a convex optimization problem which offers a numerically tractable solution to deal with control problems that may have no analytical solution. Hence, reducing a control design problem to an LMI can be considered as a practical solution to this problem (Boyd et al., 1994). Solving robust control problems by reducing to LMI problems has become a widely accepted technique (Balakrishnan & Wang, 2000). General multi objectives control problems, such as H2 and Hoo performance, peak to peak gain, passivity, regional pole placement and robust regulation are notoriously difficult, but these can be solved by formulating the problems into linear matrix inequalities (LMIs) (Boyd et al, 1994; Scherer et al, 1997)). The objective of this paper is to propose a parameter selection technique of PID controller within the framework of robust control theory with linear matrix inequalities. This is an alternative method to optimize the adjustment of a PID controller to achieve the performance limits and to determine the existence of satisfactory controllers by only using two design parameters instead of three well known parameters in the PID controller. By using optimization method, an absolute scale of merits subject to any designs can be measured. The advantage of the proposed technique is implementing an output feedback control (PID controller) by taking the simplicity in the full state feedback design. The proposed technique can be applied either to a single-input-single-output (SISO) or to a multi-inputs-multi-outputs (MIMO) PID controller. The paper is organised as follows. Section 2 describes the formulation of the PID controller in the full state feedback representation. In section 3, the synthesis of Hoo dissipative integral backstepping is applied to the PID controller using two design parameters. This section also provides a derivation of the algebraic Riccati inequality (ARI) formulation for the robust control from the dissipative integral backstepping synthesis. Section 4 illustrates an application of the robust PID controller for time delay uncertainties compensation in a network control system problem. Section 5 provides some conclusions. 2. State feedback representation of PID controller In order to design with robust control theory, the PID controller is expressed as a full state feedback control law. Consider a single input single output linear time invariant plant described by the linear differential equation Robust H. PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 71 x(t) = Ax(t) + B 2 u(t) y(t) = C 2 x(t) ' with some uncertainties in the plant which will be explained later. Here, the states x <=R n are the solution of (1), the control signal ueR 1 is assumed to be the output of a PID controller with input y e R 1 . The PID controller for regulator problem is of the form u(t) = K,\y{t)d{t) + K 2 y(t) + K 3 ±y(t) (2) which is an output feedback control system and K 1 =K p / T z , K 2 =K , K 3 = K T d of which K , T i and T d denote proportional gain, time integral and time derivative of the well known PID controller respectively. The structure in equation (2) is known as the standard PID controller (Astrom & Hagglund, 1995). The control law (2) is expressed as a state feedback law using (1) by differentiating the plant output y as follows y = C 2 x y = C 2 Ax + C 2 B 2 u y = C 2 A 2 x + C 2 AB 2 u + C 2 B 2 u This means that the derivative of the control signal (2) may be written as (1 - K 3 C 2 B 2 )u - (K 3 C 2 A 2 + K 2 C 2 A + K a C 2 )x - (K 3 C 2 AB 2 + K 2 C 2 B 2 )u = (3) Using the notation K as a normalization of K , this can be written in more compact form K = [K a K 2 K 3 ]=(l-K 3 C 2 B 2 y 1 [K 1 K 2 K 3 ] (4) or K = cK where c is a scalar. This control law is then given by u = K[Cl A T Cl (A 2 ) T Cjfx+ K[0 B T 2 C T 2 B T 2 A T C T 2 fu (5) Denote K x = K[C\ A T C T 2 (A 2 ) T Cj] T and K u = K[0 B\c\ BjA T Cj] T , the block diagram of the control law (5) is shown in Fig. 1. In the state feedback representation, it can be seen that the PID controller has interesting features. It has state feedback in the upper loop and pure integrator backstepping in the lower loop. By means of the internal model principle (IMP) (Francis & Wonham, 1976; Joelianto & Williamson, 2009), the integrator also guarantees that the PID controller will give zero tracking error for a step reference signal. Equation (5) represents an output feedback law with constrained state feedback. That is, the control signal (2) may be written as u a = K a x a (6) where x U a = U > X a = 72 Robust Control, Theory and Applications :[[c 2 T K a = K\[C T 2 A T C T 2 (A 2 ) T C T 2 ] T [0 B T 2 C\ BjA T Cjf] Arranging the equation and eliminating the transpose lead to K=K C 2 C 2 A C 2 B \^ 2 /\ \^- 2 /\d 2 --KT The augmented system equation is obtained from (1) and (7) as follows ^a= A a X a +B a U a (7) (8) where A n A B 2 ;B fl Fig. 1. Block diagram of state space representation of PID controller Equation (6), (7) and (8) show that the PID controller can be viewed as a state variable feedback law for the original system augmented with an integrator at its input. The augmented formulation also shows the same structure known as the integral backstepping method (Krstic et al., 1995) with one pure integrator. Hence, the selection of the parameters of the PID controller (6) via full state feedback gain is inherently an integral backstepping control problems. The problem of the parameters selection of the PID controller becomes an optimal problem once a performance index of the augmented system (8) is defined. The parameters of the PID controller are then obtained by solving equation (7) that requires the inversion of the matrix r . Since r is, in general, not a square matrix, a numerical method should be used to obtain the inverse. Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 73_ For the sake of simplicity, the problem has been set-up in a single-input-single-output (SISO) case. The extension of the method to a multi-inputs-multi-outputs (MIMO) case is straighforward. In MIMO PID controller, the control signal has dimension m , ueR m is assumed to be the output of a PID controller with input has dimension p , y e R v . The parameters of the PID controller K lf K 2 , and K 3 will be square matrices with appropriate dimension. 3. Hoc dissipative integral backstepping synthesis The backstepping method developed by (Krstic et al., 1995) has received considerable attention and has become a well known method for control system designs in the last decade. The backstepping design is a recursive algorithm that steps back toward the control input by means of integrations. In nonlinear control system designs, backstepping can be used to force a nonlinear system to behave like a linear system in a new set of coordinates with flexibility to avoid cancellation of useful nonlinearities and to focus on the objectives of stabilization and tracking. Here, the paper combines the advantage of the backstepping method, dissipative control and Hoc optimal control for the case of parameters selection of the PID controller to develop a new robust PID controller design. Consider the single input single output linear time invariant plant in standard form used in Hoc performance by the state space equation x(t) = Ax(t) + B^t) + B 2 u(t), x(0) = x z(t) = C a x(0 + D n w(t) + D 12 u(t) (9) y(t) = C 2 x(t) + D 21 w(t) + D 22 u(t) where x <=R n denotes the state vector, ueR 1 is the control input, w <=R P is an external input and represents driving signals that generate reference signals, disturbances, and measurement noise, y eR 1 is the plant output, and zeR m is a vector of output signals related to the performance of the control system. Definition 1. A system is dissipative (Yuliar et al., 1998) with respect to supply rate r(z(t),w(t)) for each initial condition x if there exists a storage function V , V : R n — » R + satisfies the inequality h V(x(t )) + J r(z(t), w(t))dt > V(*(* a )) , V(t a ,t )eR + ,x eR n (10) to and t < t x and all trajectories (x,y,z) which satisfies (9). The supply rate function r(z(t),w(t)) should be interpreted as the supply delivered to the h system. If in the interval [£ ^i] the integral f r(z(t),w(t))dt is positive then work has been done to the system. Otherwise work is done by the system. The supply rate determines not only the dissipativity of the system but also the required performance index of the control system. The storage function V generalizes the notion of an energy function for a dissipative system. The function characterizes the change of internal storage V(x(t 1 ))-V(x(t )) in any interval [t ,ti]/ and ensures that the change will never exceed the amount of the supply into 74 Robust Control, Theory and Applications the system. The dissipative method provides a unifying tool as index performances of control systems can be expressed in a general supply rate by selecting values of the supply rate parameters. The general quadratic supply rate function (Hill & Moylan, 1977) is given by the following equation r(z,w) = —(w T Qw + 2w T Sz + z T Rz) (11) where Q and R are symmetric matrices and Q(x) = Q + SD n (x) -D T n (x)S T - D^RD^x) such that Q(x)>0 for xeR n and R<0 and inf^ M {a min (Q(x))} = k > . This general supply rate represents general problems in control system designs by proper selection of matrices Q, R and S (Hill & Moylan, 1977; Yuliar et al, 1997): finite gain (H*,) performance ( Q = y 2 I , S = and R = -I); passivity ( Q = R = and S = I ); and mixed Hoc- positive real performance ( Q = 6y 2 I , R = -QI and S = (1-6)1 for 6e[0,l]). For the PID control problem in the robust control framework, the plant ( E ) is given by the state space equation E = x(t) = Ax(t) + B 1 w(f) + B 2 u(t),x(0) = x z(t) = (12) K D 12 u(t), with D n = and y > with the quadratic supply rate function for Hoc performance r(z,w) = —(y 2 w T w - z T z) Next the plant ( E ) is added with integral backstepping on the control input as follows x(t) = Ax(t) + B x w{t) + B 2 u(t) u a (t) = u(t) C a x(0 (13) (14) z(t) = D 12 u(t) where p is the parameter of the integral backstepping which act on the derivative of the control signal u(t) . In equation (14), the parameter p > is a tuning parameter of the PID controller in the state space representation to determine the rate of the control signal. Note that the standard PID controller in the state feedback representation in the equations (6), (7) and (8) corresponds to the integral backstepping parameter p = 1 . Note that, in this design the gains of the PID controller are replaced by two new design parameters namely y and p which correspond to the robustness of the closed loop system and the control effort. The state space representation of the plant with an integrator backstepping in equation (14) can then be written in the augmented form as follows Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 75 -±{t) u(t)_ = "A B 2 ~ ~x(t) u(t)_ + w(t) + "0" 1 u a (t) z(t) = C a D 19 x(t) u(t) "«(') The compact form of the augmented plant ( I a ) is given by Kit) = A a x a (t) + B w w(t) + B a u a (t);x a (0) = ^nO 2(f) = CA(f) + D la ro(f) + D 2 A(f) where /A B = "A B 2 " ^ w = /B B = "0" 1 ,c a = "Ci " D 12 ^D 2fl = "0" u J LpJ Now consider the full state gain feedback of the form ««(') = U(f) (15) (16) (17) The objective is then to find the gain feedback K a which stabilizes the augmented plant ( I a ) with respect to the dissipative function V in (10) by a parameter selection of the quadratic supply rate (11) for a particular control performance. Fig. 2. shows the system description of the augmented system of the plant and the integral backstepping with the state feedback control law. y ^a U a ~ K a X a w Fig. 2. System description of the augmented system 76 Robust Control, Theory and Applications The following theorem gives the existence condition and the formula of the stabilizing gain feedback K a . Theorem 2. Given y > and p > . If there exists X = X T > of the following Algebraic Riccati Inequality ~A B 2 ~A T 0~ ( p- 2 ro o~ -y" 2 + X-X L° ° J Bl V [u 1 X Then the full state feedback gain K n bX o -p- 2 B fl T X = -p- 2 [0 1]X x + clc, o dLd. \iy\i. < (18) (19) leads to | \Z\ | 00 <y Proof. Consider the standard system (9) with the full state feedback gain u(t) = Kx(t) and the closed loop system x(t) = A u x(t) + B 1 w(t), x(0) = x z(t) = C u x(t) + D n w(t) where D n = , A u =A + B 2 K , C u =C 1 + D 12 K is strictly dissipative with respect to the quadratic supply rate (11) such that the matrix A u is asymptotically stable. This implies that the related system x(t) = Ax(t) + B 1 w(t), x(0) = x z(t) = C x x{t) where A = A u - B^SC" , B 1 =B 1 Q~ 1/2 and C 1 =(S T Q~ 1 S -R) 1/2 C U has Hoo norm strictly less than 1, which implies there exits a matrix X > solving the following Algebraic Riccati Inequality (ARI) (Petersen et al. 1991) A T X + XA + XB X B\X + C\C X < In terms of the parameter of the original system, this can be written as (A U ) T X + XA U + [XB 1 -(C u ) T S T ]Q~ 1 [BlX-SC u ]- (C u ) T RC u <0 Define the full state feedback gain K = -E" 1 ((B 2 - B^SD^ ) T X + D 12J RC a (20) (21) (22) By inserting Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 77 A U =A + B 2 K, C U =C 1 +D 12 K S=S + D T 11 R, Q = Q + SD 11 +D T 11 S T +D T 11 RD 11 R = S T Q~ 1 S-R, E = D T 12 RD 12 B = B 2 - R&^SDu ,5 = 1- D 12 E _1 D[ 2 £ into (21) , completing the squares and removing the gain K give the following ARI X(A - BE _1 D[ 2 JRC a - B&SC! ) + ( A - BE^D^R^ - B^S^ )X - -X(B£- 1 B T - B^B^X + d[D T RDC 1 < (23) Using the results (Scherer, 1990), if there exists X > satisfies (23) then K given by (22) is stabilizing such that the closed loop system A u = A + B 2 K is asymptotically stable. Now consider the augmented plant with integral backstepping in (16). In this case, D lfl =[0 Of . Note that E)\ a C a = and D la = . The Hoo performance is satisfied by setting the quadratic supply rate (11) as follow: S = 0, R = -R = I, E = D T 2a RD 2a =D T 2tt D 2a , B = B a , D = 7-D 2fl (D 2 T fl D 2fl )- 1 D 2 T fl Inserting the setting to the ARI (23) yields X(A a - B (D 2 T fl D 2fl )- 1 D 2 T 7C - B w Q- 1 0C a ) + +{A a - B a {p\p 2a r x D\ a lC a - B w Q-'QC a )X - -X(B (D 2 T fl D 2fl )- 1 B: - B^B^X + +(Cj(I -D 2a (D T 2tt D 2tt y'D T 2a ) T x (I - D 2fl (D 2 T fl D 2fl )- 1 D 2 r fl )C fl ) < The equation can then be written in compact form XA a + A T a X - X(p" 2 B X - y~ 2 B w Bl )X + C T a C a < this gives (18). The full state feedback gain is then found by inserting the setting into (22) (24) K a = -E- 1 ((B fl - B w Q- 1 SD 2a ) T X- D T 2a RC a ) that gives | | 27 1 | 00 <y (Yuliar et al, 1998; Scherer & Weiland, 1999). This completes the proof. The relation of the ARI solution (8) to the ARE solution is shown in the following. Let the transfer function of the plant (9) is represented by z(s) y( s ). P n (s) P 12 (s) P 21 (s) P 22 (s) w(s) u(s) and assume the following conditions hold: (Al). (A,B 2 ,C 2 ) is stabilizable and detectable (A2). D 22 =0 78 Robust Control, Theory and Applications (A3). D 12 has full column rank, D 21 has full row rank (A4). P 12 (s) and P 2 i( s ) have no invariant zero on the imaginary axis From (Gahinet & Apkarian, 1994), equivalently the Algebraic Riccati Equation (ARE) given by the formula X( A - BE^D^RQ - B^SQ ) + ( A - BE^D^RC^ - B^SQ )X - (25) X(BE _1 B T - B^B^X + cItFrD^ = has a (unique) solution X^ > , such that A + B 2 K + B^YBlX - S^ + D 12 K)] is asymptotically stable. The characterization of feasible y using the Algebraic Riccati Inequality (ARI) in (24) and ARE in (25) is immediately where the solution of ARE ( X^ ) and ARI ( X ) satisfy < X., < X , X = X^ > (Gahinet & Apkarian, 1994). The Algebraic Riccati Inequality (24) by Schur complement implies A T tt X + XA a+ C T tt C tt XB a XB W B T a X p 2 / B T W X -y 2 I <0 (26) Ther problem is then to find X > such that the LMI given in (26) holds. The LMI problem can be solved using the method (Gahinet & Apkarian, 1994) which implies the solution of the ARI (18) (Liu & He, 2006). The parameters of the PID controller which are designed by using H^ dissipative integral backstepping can then be found by using the following algorithm: 1. Select p>0 2. Select y > 3. Find X > by solving LMI in (26) 4. Find K a using (19) 5. Find K using (7) 6. Compute K x , K 2 and K 3 using (4) 7. Apply in the PID controller (2) 8. If it is needed to achieve y minimum, repeat step 2 and 3 until y = y min then follows the next step 4. Delay time uncertainties compensation Consider the plant given by a first order system with delay time which is common assumption in industrial process control and further assume that the delay time uncertainties belongs to an a priori known interval Y(s) = -?—e- Ls U(s) , I g [a,b] (27) TS + 1 The example is taken from (Joelianto et al., 2008) which represents a problem in industrial process control due to the implementation of industrial digital data communication via Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 79 ethernet networks with fieldbus topology from the controller to the sensor and the actuator (Hops et al., 2004; Jones, 2006, Joelianto & Hosana, 2009). In order to write in the state space representation, the delay time is approximated by using the first order Pade approximation Y(s) = -ds + 1 xs + 1 ds + 1 U(s), d = L/2 (28) In this case, the values of x and d are assumed as follows: x = 1 s and d nom = 3 s. These pose a difficult problem as the ratio between the delay time and the time constant is greater than one ( (d I x) > 1 ). The delay time uncertainties are assumed in the interval d e [2,4] . The delay time uncertainty is separated from its nominal value by using linear fractional transformation (LFT) that shows a feedback connection between the nominal and the uncertainty block. 8 u e y d d Fig. 3. Separation of nominal and uncertainty using LFT The delay time uncertainties can then be represented as d = ad nom + P5 , -1 < 5 < 1 d = F u 1 P a After simplification, the delay time uncertainties of any known ranges have a linear fractional transformation (LFT) representation as shown in the following figure. 8 u e y d G tot w Fig. 4. First order system with delay time uncertainty 80 Robust Control, Theory and Applications The representation of the plant augmented with the uncertainty is G tot (s) = c r a A B a B 2 1 Q Dn D 12 c 2 D 21 D 22 J (29) The corresponding matrices in (29) are A = A xll xll u x2 1 >c x = c xll o 1 D = ^u D xl2 In order to incorporate the integral backstepping design, the plant is then augmented with an integrator as follows \ = ~A B 2 ~ = B m = Acii o B xll 1-10 roi = / 1 c = c. D, D, The problem is then to find the solution X > and y > of AR1 (18) and to compute the full state feedback gain given by u a (t) = K a x a (t) = -p- 2 ([0 1]X) x(t) u(t) which is stabilizing and leads to the infinity norm | \E\ | 00 < y . The state space representation for the nominal system is given by A. -1.6667 -0.6667 1 , C nom =[-l 0.6667] Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 81 In this representation, the performance of the closed loop system will be guaranteed for the specified delay time range with fast transient response (z). The full state feedback gain of the PID controller is given by the following equation K, K* = (l-2C 3 [-l 0.6667]) M 1 K 2 W For different y , the PID parameters and transient performances, such as: settling time ( T s ) and rise time ( T r ) are calculated by using LMI (26) and presented in Table 1. For different p but fixed y , the parameters are shown in Table 2. As comparison, the PID parameters are also computed by using the standard FL performance obtained by solving ARE in (25). The results are shown Table 3 and Table 4 for different y and different p respectively. It can be seen from Table 1 and 2 that there is no clear pattern either in the settling time or the rise time. Only Table 1 shows that decreasing y decreases the value of the three parameters. On the other hand, the calculation using ARE shows that the settling time and the rise time are decreased by reducing y or p . Table 3 shows the same result with the Table 1 when the value of y is decreased. Y P K p Ki K d Tr(8) T s 5%(s) 0.1 1 0.2111 0.1768 0.0695 10.8 12.7 0.248 1 0.3023 0.2226 0.1102 8.63 13.2 0.997 1 0.7744 0.3136 0.2944 4.44 18.8 1.27 1 10.471 0.5434 0.4090 2.59 9.27 1.7 1 13.132 0.746 0.5191 1.93 13.1 Table 1. Parameters and transient response of PID for different y with LMI Y P K p Ki K d Tr(8) T s 5% (s) 0.997 0.66 11.019 0.1064 0.3127 39.8 122 0.997 0.77 0.9469 0.2407 0.3113 13.5 39.7 0.997 1 0.7744 0.3136 0.2944 4.44 18.8 0.997 1.24 0.4855 0.1369 0.1886 21.6 56.8 0.997 1.5 0.2923 0.0350 0.1151 94.4 250 Table 2. Parameters and transient response of PID for different p with LMI 82 Robust Control, Theory and Applications Y P K p Ki K d T r (s) T s 5% (s) 0.1 1 0.2317 0.055 0.1228 55.1 143 0.248 1 0.2319 0.0551 0.123 55.0 141 0.997 1 0.2373 0.0566 0.126 53.8 138 1.27 1 0.2411 0.0577 0.128 52.6 135 1.7 1 0.2495 0.0601 0.1327 52.2 130 Table 3. Parameters and transient response of PID for different y with ARE Y P K p Ki K d Tr(s) T s 5%(s) 0.997 0.66 0.5322 0.1396 0.2879 21.9 57.6 0.997 0.77 0.4024 0.1023 0.2164 29.7 77.5 0.997 1 0.2373 0.0566 0.126 39.1 138 0.997 1.24 0.1480 0.0332 0.0777 91.0 234 0.997 1.5 0.0959 0.0202 0.0498 150.0 383 Table 4. Parameters and transient response of PID for different p with ARE -c* -1.5 \C 2C :■!.: 40 50 £0 Time (sec) 7D -o iC 1CD Fig. 5. Transient response for different y using LMI Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 83 Fig. 6. Transient response for different p using LMI -1 -0.3 -0.6 -0. -0.2 0.2 0.4 Real Axis Fig. 7. Nyquist plot y = 0.248 and p = 1 using LMI 84 Robust Control, Theory and Applications i i i _ - j>^~^"" "■---! 1 1 1 : : ' *\s --* : : / / / / s :- . / J 1 - / \ » 0.2 j \ 3 \ ■ i " \^^~~^^ "1 /^- — J 1 E -0.2 \ ( / -0.4 ' \ 1 / - \ / -0.6 \ \ ^ -03 **-%^N. ,,''* ' 1 1 1 ^=-%M.:»n, -1 -OS -06 -0.4 -0:2 0.2 04 0.6 03 1 Real A*fc Fig. 8. Nyquist plot y = 0.997 and p - 0.66 using LMI It 30 90 1W Fig. 9. Transient response for different d using LMI Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 85 SO 100 120 Time- {sec) ??.?. Fig. 10. Transient response for different bigger d using LMI The simulation results are shown in Figure 5 and 6 for LMI, with y and p are denoted by g and r respectively in the figure. The LMI method leads to faster transient response compared to the ARE method for all values of y and p . Nyquist plots in Figure 7 and 8 show that the LMI method has small gain margin. In general, it also holds for phase margin except at y = 0.997 and p = 1.5 where LMI has bigger phase margin. In order to test the robustness to the specified delay time uncertainties, the obtained robust PID controller with parameter y =0.1 and p = 1 is tested by perturbing the delay time in the range value of d e [1,4] . The results of using LMI are shown in Figure 9 and 10 respectively. The LMI method yields faster transient responses where it tends to oscillate at bigger time delay. With the same parameters y and p , the PID controller is subjected to bigger delay time than the design specification. The LMI method can handle the ratio of delay time and time constant L / x < 12 s while the ARE method has bigger ratio L / x < 43 s. In summary, simulation results showed that LMI method produced fast transient response of the closed loop system with no overshoot and the capability in handling uncertainties. If the range of the uncertainties is known, the stability and the performance of the closed loop system will be guaranteed. 5. Conclusion The paper has presented a model based method to select the optimum setting of the PID controller using robust FL dissipative integral backstepping method with state feedback synthesis. The state feedback gain is found by using LMI solution of Algebraic Riccati Inequality (ARI). The paper also derives the synthesis of the state feedback gain of robust FL dissipative integral backstepping method. The parameters of the PID controller are 86 Robust Control, Theory and Applications calculated by using two new parameters which correspond to the infinity norm and the weighting of the control signal of the closed loop system. The LMI method will guarantee the stability and the performance of the closed loop system if the range of the uncertainties is included in the LFT representation of the model. The LFT representation in the design can also be extended to include plant uncertainties, multiplicative perturbation, pole clustering, etc. Hence, the problem will be considered as multi objectives LMI based robust Hoo PID controller problem. The proposed approach can be directly extended for MIMO control problem with MIMO PID controller. 6. References Astrom, K.J. & Hagglund, T. (1995). PID Controllers: Theory, Design, and Tuning, second ed., Instrument Society of America, ISBN 1-55617-516-7, Research Triangle Park, North Carolina - USA Boyd, S.P. & Barrat, C.H. (1991). 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Non-deterministic DUT behavior during functional testing of high speed serial busses: challenges and solutions, Proceedings International Test Conference, ISBN 0-7803-8581-0, 26-28 Oct. 2004, IEEE, Charlotte, NC, USA Hwang, S.H. & Chang, H.C. (1987). A theoritical examination of closed-loop properties and tuning methods of single-loop PI controllers, Chemical Engineering Science, Vol. 42, pp. 2395-2415, ISSN 0009-2509 Robust hL PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis 87 Joelianto, E. & Tommy. (2003). A robust DC to DC buckboost converter using PID Hoo- backstepping controller, Proceedings of Int. Confer, on Power Electronics and Drive Systems (PEDS), pp. 591-594, ISBN 0-7803-7885-7, 17-20 Nov. 2003, Singapore Joelianto, E.; Sutarto, H.Y. & Wicaksono, A. (2008). Compensation of delay time uncertainties industrial control ethernet networks using LMI based robust Hoo PID controller, Proceedings of 5 th Int. Confer. 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A simple derivation of ARE solutions to the standard Hoo control problem based on LMI solution, Systems & Control Letters, Vol. 5 , pp. 487-493, ISSN 0167-6911 Petersen, I.R.; Anderson, B.D.O. & Jonkheere, E. (1991). A first principles solution to the non- singular Hoo control problem, Inter. Journal of Robust and Nonlinear Control, Vol. 1, pp. 171-185, ISSN 1099-1239 Petersen, I.R.; Ugrinovskii, V.A. & Savkin, A.V. (2000). Robust Control Design using Hoo Methods, Springer, ISBN 1-8523-3171-2, London Scherer, C. (1990). The Riccati Inequality and State-Space Hcc-Optimal Control, PhD Dissertation, Bayerischen Julius Maximilans, Universitat Wurzburg, Wurzburg Scherer, C; Gahinet, P. & Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization, IEEE Trans, on Automatic Control, Vol. 42, pp. 896-911, ISSN 0018-9286 Scherer, C. & Weiland, S. (1999). Linear Matrix Inequalities in Control, Lecture Notes DISC Course, version 2.0. http://www.er.ele.tue.nl/SWeiland/lmi99.htm Shinskey, F.G. (1996). Process Control Systems: Application, Design and Tuning, fourth ed., McGraw-Hill, ISBN 0-0705-7101-5, Boston Wang, Q.G.; Lin, C, Ye, Z., Wen, G., He, Y. & Hang, C.C. (2007). A quasi-LMI approach to computing stabilizing parameter ranges of multi-loop PID controllers, Journal of Process Control, Vol. 17, pp. 59-72, ISSN 0959-1524 Williamson, D. & Moore, J.B. (1971). Three term controller parameter selection using suboptimal regulator theory, IEEE Trans, on Automatic Control, Vol. 16, pp. 82-83, ISSN 0018-9286 Wong, S.K.P. & Seborg, D.E. (1988). Control strategy for single-input-single-output nonlinear systems with time delays, International Journal of Control, Vol. 48, pp. 2303-2327, ISSN 0020-7179 print/ISSN 1366-5820 online Yuliar, S.; James, M.R. & Helton, J.W. (1998). 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Introduction The innate immune system provides a tactical response, signaling the presence of 'non-self organisms and activating B cells to produce antibodies to bind to the intruders 7 epitopic sites. The antibodies identify targets for scavenging cells that engulf and consume the microbes, reducing them to non-functioning units (Stengel et al., 2002b). The antibodies also stimulate the production of cytokines, complement factors and acute-phase response proteins that either damage an intruder's plasma membrane directly or trigger the second phase of immune response. The innate immune system protects against many extracellular bacteria or free viruses found in blood plasma, lymph, tissue fluid, or interstitial space between cells, but it cannot clean out microbes that burrow into cells, such as viruses, intracellular bacteria, and protozoa (Janeway, 2005; Lydyard et al., 2000; Stengel et al., 2002b). The innate immune system is a complex system and the obscure relationships between the immune system and the environment in which several modulatory stimuli are embedded (e.g. antigens, molecules of various origin, physical stimuli, stress stimuli). This environment is noisy because of the great amount of such signals. The immune noise has therefore at least two components: (a) the internal noise, due to the exchange of a network of molecular and cellular signals belonging to the immune system during an immune response or in the homeostasis of the immune system. The concept of the internal noise might be viewed in biological terms as a status of sub-inflammation required by the immune response to occur; (b) the external noise, the set of external signals that target the immune system (and hence that add noise to the internal one) during the whole life of an organism. For clinical treatment of infection, several available methods focus on killing the invading microbes, neutralizing their response, and providing palliative or healing care to other organs of the body. Few biological or chemical agents have just one single effect; for example, an agent that kills a virus may also damage healthy 'self cells. A critical function of drug discovery and development is to identify new compounds that have maximum intended efficacy with minimal side effects on the general population. These examples include antibiotics as microbe killers; interferons as microbe neutralizers; interleukins, antigens from killed (i.e. non-toxic) pathogens, and pre-formed and monoclonal antibodies as immunity enhancers (each of very different nature); and anti-inflammatory and anti- histamine compounds as palliative drugs (Stengel et al., 2002b). Recently, several models of immune response to infection (Asachenkov, 1994; Nowak & May, 2000; Perelson & Weisbuch, 1997; Rundell et al., 1995) with emphasis on the human- 90 Robust Control, Theory and Applications immunodeficiency virus have been reported (Nowak et al., 1995; Perelson et al., 1993; Perelson et al, 1996; Stafford et al, 2000). Norbert Wiener (Wiener, 1948) and Richard Bellman (Bellman, 1983) appreciated and anticipated the application of mathematical analysis for treatment in a broad sense, and Swan made surveys on early optimal control applications to biomedical problems (Swan, 1981). Kirschner (Kirschner et al., 1997) offers an optimal control approach to HIV treatment, and intuitive control approaches are presented in (Bonhoeffer et al, 1997; De Boer & Boucher, 1996; Wein et al, 1998; Wodarz & Nowak, 1999, 2000). The dynamics of drug response (pharmacokinetics) are modeled in several works (Robinson, 1986; van Rossum et al., 1986) and control theory is applied to drug delivery in other studies (Bell & Katusiime, 1980; Carson et al, 1985; Chizeck & Katona, 1985; Gentilini et al, 2001; Jelliffe, 1986; Kwong et al, 1995; Parker et al, 1996; Polycarpou & Conway, 1995; Schumitzky, 1986). Recently, Stengel (Stengel et al, 2002a) presented a simple model for the response of the innate immune system to infection and therapy, reviewed the prior method and results of optimization, and introduced a significant extension to the optimal control of enhancing the immune response by solving a two-point boundary-value problem via an iterative method. Their results show that not only the progression from an initially life- threatening state to a controlled or cured condition but also the optimal history of therapeutic agents that produces that condition. In their study, the therapeutic method is extended by adding linear-optimal feedback control to the nominal optimal solution. However, the performance of quadratic optimal control for immune systems may be decayed by the continuous exogenous pathogen input, which is considered as an environmental disturbance of the immune system. Further, some overshoots may occur in the optimal control process and may lead to organ failure because the quadratic optimal control only minimizes a quadratic cost function that is only the integration of squares of states and allows the existence of overshoot (Zhou et al., 1996). Recently, a minimax control scheme of innate immune system is proposed by the dynamic game theory approach to treat the robust control with unknown disturbance and initial condition (Chen et al., 2008). They consider unknown disturbance and initial condition as a player who wants to destroy the immune system and a control scheme as another player to protect the innate immune system against the disturbance and uncertain initial condition. However, they assume that all state variables are available. It is not the case in practical application. In this study, a robust Hoo tracking control of immune response is proposed for therapeutic enhancement to track a desired immune response under stochastic exogenous pathogen input, environmental disturbances and uncertain initial states. Furthermore, the state variables may not be all available and the measurement is corrupted by noises too. Therefore, a state observer is employed for state estimation before state feedback control of stochastic immune systems. Since the statistics of these stochastic factors may be unknown or unavailable, the Hoo observer-based control methodology is employed for robust Hoo tracking design of stochastic immune systems. In order to attenuate the stochastic effects of stochastic factors on the tracking error, their effects should be considered in the stochastic Hoo tracking control procedure from the robust design perspective. The effect of all possible stochastic factors on the tracking error to a desired immune response, which is generated by a desired model, should be controlled below a prescribed level for the enhanced immune systems, i.e. the proposed robust Hoo tracking control need to be designed from the stochastic Hoo tracking perspective. Since the stochastic innate immune system is highly Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 91 nonlinear, it is not easy to solve the robust observer-based tracking control problem by the stochastic nonlinear Hoo tracking method directly. Recently, fuzzy systems have been employed to efficiently approximate nonlinear dynamic systems to efficiently treat the nonlinear control problem (Chen et aL, 1999,2000; Li et al., 2004; Lian et aL, 2001). A fuzzy model is proposed to interpolate several linearized stochastic immune systems at different operating points to approximate the nonlinear stochastic innate immune system via smooth fuzzy membership functions. Then, with the help of fuzzy approximation method, a fuzzy Hoo tracking scheme is developed so that the Hoo tracking control of stochastic nonlinear immune systems could be easily solved by interpolating a set of linear Hoo tracking systems, which can be solved by a constrained optimization scheme via the linear matrix inequality (LMI) technique (Boyd, 1994) with the help of Robust Control Toolbox in Matlab (Balas et al., 2007). Since the fuzzy dynamic model can approximate any nonlinear stochastic dynamic system, the proposed Hoo tracking method via fuzzy approximation can be applied to the robust control design of any model of nonlinear stochastic immune system that can be T-S fuzzy interpolated. Finally, a computational simulation example is given to illustrate the design procedure and to confirm the efficiency and efficacy of the proposed Hoo tracking control method for stochastic immune systems under external disturbances and measurement noises. 2. Model of innate immune response A simple four-nonlinear, ordinary differential equation for the dynamic model of infectious disease is introduced here to describe the rates of change of pathogen, immune cell and antibody concentrations and as an indicator of organic health (Asachenkov, 1994; Stengel et al., 2002a). In general, the innate immune system is corrupted by environmental noises. Further, some state variable cannot be measured directly and the state measurement may be corrupted by measurement noises. A more general dynamic model will be given in the sequel. x x - (a n - a 12 x 3 )x 1 + \u x + w 1 Xy — Cly-] \X A )t*22 1 3 23 \ 2 2 / "?2 tVy x 3 = a 31 x 2 - (a 32 + a 33 x x )x 3 + b 3 u 3 + w 3 x 4 = a A1 x t - a 42 x 4 + b 4 u 4 + w 4 (1) y 1 = c 1 x 2 +n lf y 2 = c 2 x 3 +n 2 ,y 3 = c 3 x A + n 3 fcos(7ix 4 ), 0<x 4 <0.5 a ^ ) = \ 0, 0.5<* 4 where X\ denotes the concentration of a pathogen that expresses a specific foreign antigen; xi denotes the concentration of immune cells that are specific to the foreign antigen; X3 denotes the concentration of antibodies that bind to the foreign antigen; X4 denotes the characteristic of a damaged organ [j4=0: healthy, X4 ^ 1: dead]. The combined therapeutic control agents and the exogenous inputs are described as follows: u\ denotes the pathogen killer's agent; U2 denotes the immune cell enhancer; W3 denotes the antibody enhancer; W4 denotes the organ healing factor (or health enhancer); w\ denotes the rate of continuing introduction of exogenous pathogens; wi ~ W4 denote the environmental disturbances or unmodeled errors and residues; w\ ~ w^ are zero mean white noises, whose covariances are uncertain or 92 Robust Control, Theory and Applications unavailable; and #21 (#4) is a nonlinear function that describes the mediation of immune cell generation by the damaged cell organ. And if there is no antigen, then the immune cell maintains the steady equilibrium value of xi. The parameters have been chosen to produce a system that recovers naturally from the pathogen infections (without treatment) as a function of initial conditions during a period of times. Here, y\, yz, 1/3 are the measurements of the corresponding states; c\, ci, C3 are the measurement scales; and rt\, n% H3 are the measurement noises. In this study, we assume the measurement of pathogen x\ is unavailable. For the benchmark example in (1), both parameters and time units are abstractions, as no specific disease is addressed. The state and control are always positive because concentrations cannot go below zero, and organ death is indicated when X4 > 1. The structural relationship of system variables in (1) is illustrated in Fig. 1. Organ health mediates immune cell production, inferring a relationship between immune response and fitness of the individual. Antibodies bind to the attacking antigens, thereby killing pathogenic microbes directly, activating complement proteins, or triggering an attack by phagocytic cells, e.g. macrophages and neutrophils. Each element of the state is subject to an independent control, and new microbes may continue to enter the system. In reality, however, the concentration of invaded pathogens is hardly to be measured. We assume that only the rest of three elements can be measured with measurement noises by medical devices or other biological techniques such as an immunofluorescence microscope, which is a technique based on the ability of antibodies to recognize and bind to specific molecules. It is then possible to detect the number of molecules easily by using a fluorescence microscope (Piston, 1999). Exogenous pathogens Wi ^•« Environmental disturbances w(t) n 3 1 — -w*"- _ ► t y/Measurement noise n(t) Fig. 1. Innate and enhanced immune response to a pathogenic attack under exogenous pathogens, environmental disturbances, and measurement noises. Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 93 Several typical uncontrolled responses to increasing levels of initial pathogen concentration under sub-clinical, clinical, chronic, and lethal conditions have been discussed and shown in Fig. 2 (Stengel et aL, 2002a). In general, the sub-clinical response would not require medical examination, while the clinical case warrants medical consultation but is self-healing without intervention. Pathogen concentration stabilizes at non-zero values in the chronic case, which is characterized by permanently degraded organ health, and pathogen concentration diverges without treatment in the lethal case and kills the organ (Stengel et aL, 2002b). Finally, a more general disease dynamic model for immune response could be represented as x(t) = f(x(t)) + g(x(t))u(t) + Dw(t), y(t) = c(x(t)) + n(t) *(0) = (2) where x(t) e R nxl is the state vector; u(t) e R is the control agent; w(t) e R includes exogenous pathogens, environmental disturbances or model uncertainty. y(t) e R /xl is the measurement output; and n(t) e R /xl is the measurement noises. We assume that w(t) and n(t) are independent stochastic noises, whose covariances may be uncertain or unavailable. All possible nonlinear interactions in the immune system are represented by f(x(t)). sub-clinical clinical Time unit chronic AAAAAAAAAAA « fl - oooooooo 000 o 4 6 Time unit lethal Fig. 2. Native immune responses to attack by different pathogens which are sub-clinical, clinical, chronic, and lethal conditions (Stengel et aL, 2002a). 94 Robust Control, Theory and Applications 3. Robust Hoc Therapeutic Control of Stochastic Innate Immune Response Our control design purpose for nonlinear stochastic innate immune system in (2) is to specify a state feedback control u(t) = k(x(t) - x d (t)) so that the immune system can track the desired response Xd(t). Since the state variables are unavailable for feedback tracking control, the state variables have to be estimated for feedback tracking control u(t) = k(x(t)-x d (t)) . Suppose the following observer-based control with y(t) as input and u(t) as output is proposed for robust Hoc tracking control. x(t) = f(x(t)) + *(*(*))«(') + Kx(t))(y(t) - c(x(t))) u(t) = k(x(t)-x d (t)) (3) where the observer-gain l(x(t )) is to be specified so that the estimation error e(t) = x(t) - x(t) can be as small as possible and control gain k(x(t)-x d (t)) is to be specified so that the system states x(t) can come close to the desired state responses Xd(t) from the stochastic point of view. Consider a reference model of immune system with a desired time response described as xAt) = A d x d (t) + r(t) (4) where x d (t) e R nxl is the reference state vector; A d e R nxn is a specific asymptotically stable matrix and r(t) is a desired reference signal. It is assumed that Xd(t), V£ > represents a desired immune response for nonlinear stochastic immune system in (2) to follow, i.e. the therapeutic control is to specify the observer-based control in (3) such that the tracking error x(t) = x(t)-x d (t) must be as small as possible under the influence of uncertain exogenous pathogens and environmental disturbances w(t) and measurement noises n(t). Since the measurement noises n(t), the exogenous pathogens and environmental disturbances w(t) are uncertain and the reference signal r(t) could be arbitrarily assigned, the robust Hoc tracking control design in (3) should be specified so that the stochastic effect of three uncertainties w(t), n(t) and r(t) on the tracking error could be set below a prescribed value p 2 , i.e. both the stochastic Hoc reference tracking and Hoc state estimation should be achieved simultaneously under uncertain w(t),n(t) and r(t). f f (x T (t)Q,m + e T (t)Q 2 e(t))dt f f/ (w T (t)w(t) + n T (t)n(t) + r T (t)r(t))dt JO (5) where the weighting matrices Q; are assumed to be diagonal as follows Qi q l n q 22 fe <?44 z = l,2. The diagonal element q J u of Qi denotes the punishment on the corresponding tracking error and estimation error. Since the stochastic effect of w(t), r(t) and n(t) on tracking error x(t) Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 95 and estimation error e(t) is prescribed below a desired attenuation level p from the energy point of view, the robust Hoc stochastic tracking problem of equation (5) is suitable for the robust Hx, stochastic tracking problem under environmental disturbances w(t), measurement noises n(t) and changeable reference r(t), which are always met in practical design cases. Remark 1: If the environmental disturbances w(t) and measurement noises n(t) are deterministic signals, the expectative symbol E[»] in (5) can be omitted. e(t)' x(t) k/(')J Let us denote the augmented vector x augmented stochastic system as then we get the dynamic equation of the x(t) = e(t) ■ x(t) 'fix) - fix) + k(x - x d )igix) - g(x)) + l(x)(c(x) - c(x)j f(x) + k(x-x d )g{x) A d x d "I 0" [*(*)] + D w(t) I lm\ The augmented stochastic system above can be represented in a general form by x(t) = F(x(t)) + Dw(t) ~f( x ) ~ f(x) + k ( x ~ x d)(g( x ) ~ g( x )) + K X M X ) ~ c ( x )) f(x) + k(x-x d )g(x) A d x d The robust H*, stochastic tracking performance in (5) can be represented by whei re F(x(t)) = ~I 0" D = D I w(t)- n(t) w(t) r(t) \ tf x T (t)Qx(t)dt f f w T (t)w(t)dt < p 2 if x(0) = or E f f x T (t)Qx(t)dt <p 2 E j tf w T (t)w(t)dt where Q - Qi o o (6) (7) and (8) Qi -Qi -Qx Q 1 _ If the stochastic initial condition x(0) *■ and is also considered in the He tracking performance, then the above stochastic H„ inequality should be modified as J o f/ x T it)Qxit)dt\ < E[V(x(0))] + p 2 E\f o f w T it)wit)dt (9) 96 Robust Control, Theory and Applications for some positive function V(x(0)) . Then we get the following result. Theorem 1: If we can specify the control gain k(x-x d ) and observer gain l(x) in the observer-based control law in (3) for stochastic immune system (2) such that the following HJI has a positive solution V(x(t)) > x(t) T Qx(t) + dx(t) \T 4p 2 dx(t) dV(x) ^ dx(t) ) < (10) Then the robust stochastic Hoc tracking performance in (5) is achieved for a prescribed tracking performance p 2 . Proof: see Appendix A. Since p 2 is a prescribed noise attenuation level of Hoo tracking performance in (5), based on the analysis above, the optimal stochastic Hoo tracking performance still need to minimize p 2 as follows Po= ™ n n P 2 (11) V(x)>0 subject to V(x(t))>0 and equation (10). At present, there does not exist any analytic or numerical solution for (10) or (11) except in very simple cases. 4. Robust fuzzy observer-based tracking control design for stochastic innate immune system Because it is very difficult to solve the nonlinear HJI in (10), no simple approach is available to solve the constrained optimization problem in (11) for robust model tracking control of stochastic innate immune system. Recently, the fuzzy T-S model has been widely applied to approximate the nonlinear system via interpolating several linearized systems at different operating points (Chen et al., 1999,2000; Takagi & Sugeno, 1985). Using fuzzy interpolation approach, the HJI in (10) can be replaced by a set of linear matrix inequalities (LMIs). In this situation, the nonlinear stochastic Hoo tracking problem in (5) could be easily solved by fuzzy method for the design of robust Hoo tracking control for stochastic innate immune response systems. Suppose the nonlinear stochastic immune system in (1) can be represented by the Takagi- Sugeno (T-S) fuzzy model (Takagi & Sugeno, 1985). The T-S fuzzy model is a piecewise interpolation of several linearized models through membership functions. The fuzzy model is described by fuzzy If-Then rules and will be employed to deal with the nonlinear Hoo tracking problem by fuzzy observer-based control to achieve a desired immune response under stochastic noises. The z-th rule of fuzzy model for nonlinear stochastic immune system in (1) is in the following form (Chen et al., 1999,2000). Plant Rule i: If z x (i) is F n and ... and z At) is F ig , then x(t) = A i x(t) + B i u(t) + Dw(t), i - 1,2,3,- •• ,L y(t) = C iX (t) + n(t) (12) Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 97 in which R is the fuzzy set; A z , B z , and C z are known constant matrices; L is the number of If-Then rules; g is the number of premise variables; and z 1 (t),z 2 (t),...,z g (t) are the premise variables. The fuzzy system is inferred as follows (Chen et al., 1999,2000; Takagi & Sugeno,1985) = X/z f (z(0)[A f x(0 + B f u(*) + Dw(t)] (13) where [ii ( z (t)) = f[F ij (z j (t)) / h i (z(t)) = -g^ , z(t) = [z 1 (t),z 2 (t),...,z g (t)] , and R(z(£)) is the grade of membership of z ; (£) in R . We assume u,(z(£))>0 and^ h .(z(f))>0 (14) Therefore, we get L I /z z (z(0)>0 and£/z f (z(0) = l (15) i=l The T-S fuzzy model in (13) is to interpolate L stochastic linear systems to approximate the nonlinear system in (1) via the fuzzy basis functions h ( (z(t)) . We could specify the L L parameter A z and B- easily so that ^fy(z(£))A-x(£) and ^h i (z(t))B i in (13) can approximate F(x(t)) and g(x(t)) in (2) by the fuzzy identification method (Takagi & Sugeno, 1985). By using fuzzy If-Then rules interpolation, the fuzzy observer is proposed to deal with the state estimation of nonlinear stochastic immune system (1). Observer Rule i: If z x (i) is F n and ... and zJt) is jR, then x(t) = A i x(t) + B i u(t) + L i (y(t)-y(t)) / i = 1,2,3,—, L (16) where L; is the observer gain for the zth observer rule and y(t) = ^ l _ 1 h i (z(t))C i x(t) . The overall fuzzy observer in (16) can be represented as (Chen et al., 1999,2000) i(f ) = Yjx x (z(t ))[A t x(t ) + B,.«(t ) + L, (y(t) - y(t ))] (17) i=l 98 Robust Control, Theory and Applications Suppose the following fuzzy observer-based controller is employed to deal with the above robust Hoc tracking control design Control Rule /: If z x (i) is F a and ... and z At) is F , then M = 2> ; .(z(0)K ; (x(f)-* d (f)) 7=1 (18) Remark 2: 1. The premise variables z(£) can be measurable stable variables, outputs or combination of measurable state variables (Ma et al., 1998; Tanaka et al, 1998; Wang et al, 1996). The limitation of this approach is that some state variables must be measurable to construct the fuzzy observer and fuzzy controller. This is a common limitation for control system design of T-S fuzzy approach (Ma et al., 1998; Tanaka et al., 1998). If the premise variables of the fuzzy observer depend on the estimated state variables, i.e., z(t) instead of z(t) in the fuzzy observer, the situation becomes more complicated. In this case, it is difficult to directly find control gains Kj and observer gains U. The problem has been discussed in (Tanaka et al, 1998). 2. The problem of constructing T-S fuzzy model for nonlinear systems can be found in (Kim et al, 1997; Sugeno & Kang, 1988). Let us denote the estimation errors as e(t) = x(t)-x(t) . The estimation errors dynamic is represented as e(t) = x(t)-x(t) After manipulation, the augmented system in (6) can be expressed as the following fuzzy approximation form L I i=l Ht) = 2>,(z(f))Zfc 7 (z(f))[A, y x(f) + Ei w(t)] ;=1 (19) where A„ A,-L ; C, -B..K; A,+B,K ; . -B,K ; A, ,m- e(t) n(t) -L, D x(t) , w(t) = w(t) >E,= D x d (t) r(t) I Theorem 2: In the nonlinear stochastic immune system of (2), if P = P T > is the common solution of the following matrix inequalities: AiP + PAg- \pE i EjP + Q<Q, i,;=l,2,-,L (20) then the robust Hoo tracking control performance in (8) or (9) is guaranteed for a prescribed p 2 Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 99 In the above robust Hoo tracking control design, we don't need the statistics of disturbances, measurement noises and initial condition. We only need to eliminate their effect on the tracking error and state estimation error below a prescribed level p 2 . To obtain the best Hoo tracking performance, the optimal Hoo tracking control problem can be formulated as the following minimization problem. Po=^P 2 (21) subject to P > and equation (20). Proof: see Appendix B. In general, it is not easy to analytically determine the constrained optimization problem in (21). Fortunately, the optimal Hoo tracking control problem in (21) can be transferred into a minimization problem subject to some linear matrix inequalities (LMIs). The LMIP can be solved by a computationally efficient method using a convex optimization technique (Boyd, 1994) as described in the following. By the Schur complements (Boyd, 1994), equation (20) is equivalent to A^P + PA^ + Q FP PL -p^HH 7 )- 1 <0 (22) where L = \ Li I 0" I I and H = -I 0" D I For the convenience of design, we assume P = obtain P 22 £33. and substitute it into (22) to j 21 ± n D 12 s 22 D ?>2 ±22 ^23 S33 -P 2 I ±22 -p^DD 7 )- 1 £33 -P 2 I <0 (23) where S v - AfPn + Pu A, -CjzJ-Z t C j+ Q 2 -- S 21 = -P 22 B,K ; . S 22 = (A, + B,K ; ) T P 22 + P 22 (A ; + B,K ; ) + Q, -P 22 B,K ; . Qi S33 = AjP 33 + P 33 A d + Q 1 and Z t = P^ 100 Robust Control, Theory and Applications Since five parameters Pn, P22, P33, Kj, and U should be determined from (23) and they are highly coupled, there are no effective algorithms for solving them simultaneously till now. In the following, a decoupled method (Tseng, 2008) is provided to solve these parameters simultaneously. Note that (23) can be decoupled as Pll ^32 P 22 ^33 z l -P 2 I -p^DD 7 )- 1 ^33 -P 2 I -Z i C + Q 2 +yP 22 Z T P11 -Yi' + Qi -Qi Zr Pll -Qi AjPga + PgaAi +Qi+r^22 ^33 -P 2 I -p^DD 7 )" 1 +yP 22 Pp,?, -P 2 (24) "7^22 (-P 22 B l K j ) T -P 22 B^ (A 2+ B^ ; ) T P 2 2 + P 22 (A i+ B i K j ) + y 1 I (-^2 2 B^ ; ) r P 22 where y and y 1 are some positive scalars. Lemma 1: If -P„B,JK, "7^22 "7^22 a 23 <0 (25) Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 101 and then w 33 b 21 u 12 b 22 hi b 42 ^23 ^33 u 24 <0 u 21 ^32 hi ^33 ^24 <0 Proof: see Appendix C. From the above lemma, it is obvious that if and At-CJZj + yP22 *i p n -Yii + Qi -Qi _q Ai P 33 + P 33 A d +Qi+yP 2 i £33 Z T -P 2 I Pll -p'PD 7 )- 1 +yP 22 P33 -P 2 I -7^22 -P 22 B ! K ; . (-P 22 B,K ; ) T (A,+B,K ; ) T P 22 +P 22 (A,+B,K,) + yi I -P 22 B f K ; P 22 <0 (-P 22 B,K ; ) T -y^22 P 22 -yP 22 _ (26) (27) <0 (28) (29) then (23) holds. Remark 3: Note that (28) is related to the observer part (i.e., the parameters are Pn, P22, P33, and U) and (29) is related to the controller part (i.e., the parameters are P22 and Kj), respectively. Although the parameters P22, Kj and y are coupled nonlinearly, seven parameters Pn, P22, P33, Kj, U, y and y 1 can be determined by the following arrangement. 102 Robust Control, Theory and Applications Note that, by the Schur complements (Boyd, 1994) equation (28) is equivalent to A, Pn + PnA, -C T Z T - -Z { C + Qi z { - Yl I + Qi -Qi -Qi AjP 33+^Af+Ql ° zj -p 2 i p n P33 ° I 1 P n I £33 I -p'PD 7 )- 1 I <0 -9 2 I -y - a w 22 -y - a w 22 I -y - a w 22 _ W„: = P™ 1 , and eqi lation OS )) is eq uivc dent to (30) (-B,.Y y ) T W 22 Aj + A,W„ + Y, T Bf + B ; Y ; (-B f y ; ) T w 22 w 22 -B.-Y; w 22 w 22 yW 22 -yW 22 -Ya 1 !. <0 (31) where Y j = K ; W 22 . Therefore, if (30) and (31) are all held then (23) holds. Recall that the attenuation p can be minimized so that the optimal Hoc tracking performance in (21) is reduced to the following constrained optimization problem. Po= min p 2 (32) subject to P n > , P 22 > , P 33 > , y > , y 1 > and (30)-(31). which can be solved by decreasing p 2 as small as possible until the parameters P n > , P 22 > , P33 > , y > and y 1 > do not exist. Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 103 Remark 4: Note that the optimal Hoo tracking control problem in (32) is not a strict LMI problem since it is still a bilinear form in (30)-(31) of two scalars y and y 1 and becomes a standard linear matrix inequality problem (LMIP) (Boyd, 1994) if y and y 1 are given in advance. The decoupled method (Tseng, 2008) bring some conservatism in controller design. However, the parameters P n , P 22 = W^ , P 33 , K = YjW^ and L z = P{iZ { can be determined simultaneously from (32) by the decoupled method if scalars y and y 1 are given in advance. The useful software packages such as Robust Control Toolbox in Matlab (Balas et al., 2007) can be employed to solve the LMIP in (32) easily. In general, it is quite easy to determine scalars y and y 1 beforehand to solve the LMIP with a smaller attenuation level p 2 . In this study, the genetic algorithm (GA) is proposed to deal with the optimal Hoo tracking control problem in (32) since GA, which can simultaneously evaluate many points in the parameters space, is a very powerful searching algorithm based on the mechanics of natural selection and natural genetics. More details about GA can be found in (Jang et al, 1997). According to the analysis above, the Hoo tracking control of stochastic innate immune system via fuzzy observer-based state feedback is summarized as follows and the structural diagram of robust fuzzy observer-based tracking control design has shown in Fig. 3. Desired immune response Solving LMIs Kj X Fuzzy observer-based controller L u=Y h.{z)K.{x-x d ) T-S fuzzy model x = Y h t [A x +B i u +Dw] Nonlinear immune system x = f(x) + g(x)u +Dw Fuzzy observer x = Y hXA x +B i u +L t(y -y)] c n / u Solving LMIs Fig. 3. Structural diagram of robust fuzzy observer-based tracking control design. 104 Robust Control, Theory and Applications Design Procedure: 1. Provide a desired reference model in (4) of the immune system. 2. Select membership functions and construct fuzzy plant rules in (12). 3. Generate randomly a population of binary strings: With the binary coding method, the scalars y and y 1 would be coded as binary strings. Then solve the LMIP in (32) with scalars y and y 1 corresponding to binary string using Robust Control Toolbox in Matlab by searching the minimal value of p 2 . If the LMIP is infeasible for the corresponding string, this string is escaped from the current generation. 4. Calculate the fitness value for each passed string: In this step, the fitness value is calculated based on the attenuation level p 2 . 5. Create offspring strings to form a new generation by some simple GA operators like reproduction, crossover, and mutation: In this step, (i) strings are selected in a mating pool from the passed strings with probabilities proportional to their fitness values, (ii) and then crossover process is applied with a probability equal to a prescribed crossover rate, (iii) and finally mutation process is applied with a probability equal to a prescribed mutation rate. Repeating (i) to (iii) until enough strings are generated to form the next generation. 6. Repeat Step 3 to Step 5 for several generations until a stop criterion is met. 7. Based on the scalars y and y 1 obtained from above steps, one can obtain the attenuation level p 2 and the corresponding P n , P 22 = W 22 , P 33 , K = Y:W 22 and L { = P^Zi , simultaneously. 8. Construct the fuzzy observer in (17) and fuzzy controller in (18). 5. Computational simulation example Parameter Value Description a n 1 Pathogens reproduction rate coefficient a 12 1 The suppression by pathogens coefficient a 22 3 Immune reactivity coefficient a 23 1 The mean immune cell production rate coefficient x* 2 2 The steady-state concentration of immune cells a 31 1 Antibodies production rate coefficient a 32 1.5 The antibody mortality coefficient a 33 0.5 The rate of antibodies suppress pathogens a 41 0.5 The organ damage depends on the pathogens damage possibilities coefficient a 42 1 Organ recovery rate b-y -1 Pathogen killer's agent coefficient b 2 1 Immune cell enhancer coefficient b 3 1 Antibody enhancer coefficient fc 4 -1 Organ health enhancer coefficient c x 1 Immune cell measurement coefficient c 2 1 Antibody measurement coefficient c^ 1 Organ health measurement coefficient Table 1. Model parameters of innate immune system (Marchuk, 1983; Stengel et al., 2002b). Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 105 We consider the nonlinear stochastic innate immune system in (1), which is shown in Fig. 1. The values of the parameters are shown in Table 1. The stochastic noises of immune systems are mainly due to measurement errors, modeling errors and process noises (Milutinovic & De Boer, 2007). The rate of continuing introduction of exogenous pathogen and environmental disturbances w 1 ~ w 4 are unknown but bounded signals. Under infectious situation, the microbes infect the organ not only by an initial concentration of pathogen at the beginning but also by the continuous exogenous pathogens invasion w 1 and other environmental disturbances w 2 ~ iv 4 . In reality, however, the concentration of invaded pathogens is hardly to be measured. So, we assume that only immune cell, antibody, and organ health can be measured with measurement noises by medical devices or other biological techniques (e.g. immunofluorescence microscope). And then we can detect the numbers of molecules easily by using a fluorescence microscope (Piston, 1999). The dynamic model of stochastic innate immune system under uncertain initial states, environmental disturbances and measurement noises is controlled by a combined therapeutic control as x x = (1 - x 3 )x 1 -u 1 +w 1 x 2 = 3a 21 (x 4 )x 1 x 3 - (x 2 -2)-u 2 +w 2 a 4 ■ x 2 - (1.5 + 0.5x a )x 3 + u 3 -- 0.5x a - x 4 + w 4 + w 4 -Wo, (33) ■- X, + Tin , Vn = X A + Ho fl 21 (* 4 ) x 2 +n 1 ,y 2 =x 3 +n 2 ,y 3 -* 4: cos(7ix 4 ), 0<x 4 <0.5 0, 0.5 < x 4 A set of initial condition is assumed x(0) = [3.5 2 1.33 0] . For the convenience of simulation, we assume that w x ~ w 4 are zero mean white noises with standard deviations being all equal to 2. The measurement noises n x ~ n 3 are zero mean white noises with standard deviations being equal to 0.1. In this example, therapeutic controls u x ~ w 4 are combined to enhance the immune system. The measurable state variables y 1 ~ y 3 with measurement noises by medical devices or biological techniques are shown in Fig. 4. Our reference model design objective is that the system matrix A d and r(t) should be specified beforehand so that its transient responses and steady state of reference system for stochastic innate immune response system are desired. If the real parts of eigenvalues of A d are more negative (i.e. more robust stable), the tracking system will be more robust to the environmental disturbances. After some numerical simulations for clinical treatment, the desired reference signals are obtained by the following reference model, which is shown in Fig. 5. **(*) = -1.1 0-200 0-40 -1.5 X d(t) + B d u step(t) (34) where B d =[0 4 16/3 0] and u st (t) is the unit step function. The initial condition is given by x d (0) = [2.5 3 1.1 0.8] T . 106 Robust Control, Theory and Applications Immune cell measurement y 1 2.5 2 1.5 1 0.5 ,#T 2 3 4 5 6 Antibody measurement y 2 IW^JS ^^i^mj^ 1 2 3 4 5 Organ health measurement y 3 3 4 5 Time unit Fig. 4. The measurable state variables y 1 ~ y 3 with measurement noises n x ~ n 3 by medical devices or biological technique. 3 O o O 1 i A Time Responses of Reference Model _g — x d1 Pathogens ,_A-_ x,„ Immune cells ,_^__ x.„ Antibodies -©— x d4 Organ ^-0-^-0— 0—^--^—^— 0—^—0— 0- »-»-»-—«> °— ie— ■ o.-^ h-g-r- ^^ □ — a- — h_ 3 4 5 Time unit Fig. 5. The desired reference model with four desired states in (34): pathogens (x dl , blue, dashed square line), immune cells ( x d2 , green, dashed triangle line), antibodies ( x d3 , red, dashed diamond line) and organ ( x M , magenta, dashed, circle line) We consider the lethal case of uncontrolled stochastic immune system in Fig. 6. The pathogen concentration increases rapidly causing organ failure. We aim at curing the organ before the organ health index excesses one after a period of pathogens infection. As shown in Fig. 6, the black dashed line is a proper time to administrate drugs. Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 107 The lethal case 3 4 Time unit Fig. 6. The uncontrolled stochastic immune responses (lethal case) in (33) are shown to increase the level of pathogen concentration at the beginning of the time period. In this case, we try to administrate a treatment after a short period of pathogens infection. The cutting line (black dashed line) is an optimal time point to give drugs. The organ will survive or fail based on the organ health threshold (horizontal dotted line) [x±<l: survival, X4>1: failure]. To minimize the design effort and complexity for this nonlinear innate immune system in (33), we employ the T-S fuzzy model to construct fuzzy rules to approximate nonlinear immune system with the measurement output y 3 and y 4 as premise variables. Plant Rule i: If y 3 is F n and y 4 is F i2 , then x(t) = A ( x(t) + Bu(t) + Dw(t), i = 1,2,3,--, L y(t) = Cx(t) + n(t) To construct the fuzzy model, we must find the operating points of innate immune response. Suppose the operating points for y 3 are at y 31 = -0.333 , y 32 = 1.667 , and y 33 = 3.667 . Similarly, the operating points for y 4 are at y 41 = , y 42 = 1 , and y 43 = 2 . For the convenience, we can create three triangle-type membership functions for the two premise variables as in Fig. 7 at the operating points and the number of fuzzy rules is L = 9 . Then, we can find the fuzzy linear model parameters A z in the Appendix D as well as other parameters B , C and D . In order to accomplish the robust FL tracking performance, we should adjust a set of weighting matrices Q 1 and Q 2 in (8) or (9) as Qi 0.01 0.01 0.01 0.01 Qi 0.01 0.01 0.01 0.01 After specifying the desired reference model, we need to solve the constrained optimization problem in (32) by employing Matlab Robust Control Toolbox. Finally, we obtain the feasible parameters y = 40 and y 1 = 0.02 , and a minimum attenuation level pQ = 0.93 and a 108 Robust Control, Theory and Applications common positive-definite symmetric matrix P with diagonal matrices P n , P 22 and P 33 as follows " 0.23193 -1.5549e-4 0.083357 -0.2704 Pu = -1.5549e-4 0.010373 -1.4534e-3 -7.0637e-3 0.083357 -1.4534e-3 0.33365 0.24439 -0.2704 -7.0637e-3 0.24439 0.76177 "0.0023082 9.4449e-6 -5.7416e-5 -5.0375e-6" P 22 = 9.4449e-6 0.0016734 2.4164e-5 -1.8316e-6 -5.7416e-5 2.4164e-5 0.0015303 5.8989e-6 -5.0375e-6 -1.8316e-6 5.8989e-6 0.0015453 1.0671 -1.0849e-5 3.4209e-5 5.9619e-6 ~ ^33 = -1.0849e-5 1.9466 -1.4584e-5 1.9167e-6 3.4209e-5 -1.4584e-5 3.8941 -3.2938e-6 5.9619e-6 1.9167e-6 -3.2938e-6 1.4591 The control gain K- and the observer gain L z can also be solved in the Appendix D. ► *3 Fig. 7. Membership functions for two premise variables y 3 and y 4 . Figures 8-9 present the robust FL tracking control of stochastic immune system under the continuous exogenous pathogens, environmental disturbances and measurement noises. Figure 8 shows the responses of the uncontrolled stochastic immune system under the initial concentrations of the pathogens infection. After the one time unit (the black dashed line), we try to provide a treatment by the robust FL tracking control of pathogens infection. It is seen that the stochastic immune system approaches to the desired reference model quickly. From the simulation results, the tracking performance of the robust model tracking control via T-S fuzzy interpolation is quite satisfactory except for pathogens state x\ because the pathogens concentration cannot be measured. But, after treatment for a specific period, the pathogens are still under control. Figure 9 shows the four combined therapeutic control agents. The performance of robust FL tracking control is estimated as l' f (i T (t)Qim + e T (t)Q 2 e(t))dt \ t >{w T (t)w(t) + n T (t)n(t) + r T (t)r{t))dt -. 0.033 <p 2 = 0.93 Robust hL Tracking Control of Stochastic Innate Immune System Under Noises 109 Robust Hoo tracking control g x 1 Pathogens Take drugs ^ x„ Immune cells G— x 4 Organ Reference response 4 5 6 7 Time unit Fig. 8. The robust Hoo tracking control of stochastic immune system under the continuous exogenous pathogens, environmental disturbances and measurement noises. We try to administrate a treatment after a short period (one time unit) of pathogens infection then the stochastic immune system approach to the desired reference model quickly except for pathogens state x\. Control Agents Time unit Fig. 9. The robust Hoo tracking control in the simulation example. The drug control agents u x (blue, solid square line) for pathogens, u 2 for immune cells (green, solid triangle line), u 3 for antibodies (red, solid diamond line) and u 4 for organ (magenta, solid circle line). Obviously, the robust Hoo tracking performance is satisfied. The conservative results are due to the inherent conservation of solving LMI in (30)-(32). 110 Robust Control, Theory and Applications 6. Discussion and conclusion In this study, we have developed a robust Hoo tracking control design of stochastic immune response for therapeutic enhancement to track a prescribed immune response under uncertain initial states, environmental disturbances and measurement noises. Although the mathematical model of stochastic innate immune system is taken from the literature, it still needs to compare quantitatively with empirical evidence in practical application. For practical implementation, accurate biodynamic models are required for treatment application. However, model identification is not the topic of this paper. Furthermore, we assume that not all state variables can be measured. In the measurement process, the measured states are corrupted by noises. In this study, the statistic of disturbances, measurement noises and initial condition are assumed unavailable and cannot be used for the optimal stochastic tracking design. Therefore, the proposed Hoo observer design is employed to attenuate these measurement noises to robustly estimate the state variables for therapeutic control and Hoo control design is employed to attenuate disturbances to robustly track the desired time response of stochastic immune system simultaneity. Since the proposed Hoo observer-based tracking control design can provide an efficient way to create a real time therapeutic regime despite disturbances, measurement noises and initial condition to protect suspected patients from the pathogens infection, in the future, we will focus on applications of robust Hoo observer-based control design to therapy and drug design incorporating nanotechnology and metabolic engineering scheme. Robustness is a significant property that allows for the stochastic innate immune system to maintain its function despite exogenous pathogens, environmental disturbances, system uncertainties and measurement noises. In general, the robust Hooobserver-based tracking control design for stochastic innate immune system needs to solve a complex nonlinear Hamilton-Jacobi inequality (HJI), which is generally difficult to solve for this control design. Based on the proposed fuzzy interpolation approach, the design of nonlinear robust Hoo observer-based tracking control problem for stochastic innate immune system is transformed to solve a set of equivalent linear Hoo observer-based tracking problem. Such transformation can then provide an easier approach by solving an LMI-constrained optimization problem for robust Hoo observer-based tracking control design. With the help of the Robust Control Toolbox in Matlab instead of the HJI, we could solve these problems for robust Hoo observer-based tracking control of stochastic innate immune system more efficiently. From the in silico simulation examples, the proposed robust Hoo observer-based tracking control of stochastic immune system could track the prescribed reference time response robustly, which may lead to potential application in therapeutic drug design for a desired immune response during an infection episode. 7. Appendix 7.1 Appendix A: Proof of Theorem 1 Before the proof of Theorem 1, the following lemma is necessary. Lemma 2: For all vectors a, (3 e R nxl , the following inequality always holds a T (3 + (3 T a < — a T a + p 2 (3 T p for any scale value p > . P Let us denote a Lyapunov energy function V(x(t)) > . Consider the following equivalent equation: Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 111 £' x T (t)Qx(t)dt\ = E[V(x(0))] - E[V(z(oo))] + E f (x T (t)Qx(t) + ^|^1 ^ By the chain rule, we get dVW)) . ax(o J ^ ~[ ax(o (F(x(0) + D^(0) Substituting the above equation into (Al), by the fact that V(x(oo)) > , we get E[{V(t)Qx(0*]<E[V(x(0))] + E By Lemma 2, we have f T { 3x(t) ) dt (Al) (A2) (A3) mm) dx(t) w 2^ Dx(t) ) w 2 w Sx(f) ±P \ 5x(f) J <3x(f) (A4) Therefore, we can obtain J o V(f)Qx(Orffl<E[V(x(0))] + E f' Jo V(^(f )+ f^™Tw)) 4 P 2 mm) {. 8x(t) dx(t) ) 55T 8v<mi +p2 _ T{t)m dx(t) dt (A5) By the inequality in (10), then we get J"' x T mx(t)dt] < E[V(x(0))] + p^ff w T (t)w(t)dt (A6) If x(0) = , then we get the inequality in (8). 7.2 Appendix B: Proof of Theorem 2 Let us choose a Lyapunov energy func equation (Al) is equivalent to the following: Let us choose a Lyapunov energy function V(x(t)) = x T (t)Px(t) > where P = P T > . Then f x T (f)Qx(f)*l = E[V(x(0))] -E[V(x(»))] + Eff (x T (f)Qx(f) + 2x T (f)Px(f)) dt :E[V(x(0))] + E = E[V(x(0))] + E Jo x T (t)Qx(t) + 2x T (t)P L X^W0)Z^W0)[a 2; x(0 + e^(0] df (A7) f x T (0Qx(0 + Z^W0)Z^W0)[2x T (0PA 2; x(f) + 2x T (0PE^(0] df 112 Robust Control, Theory and Applications By Lemma 2, we have 2 *< mm) = *' mm) + & w! m) < j_^ {t)Pmm + p2 - T{t) - {t) (A8) P Therefore, we can obtain + \x T (t)PE i EjPx(t) 7=1 + p 2 w T (*)w(f) |<ft = E[V(*(0))] + E (A9) V < =1 -/ =1 + -^PE i E i i P x(t) + p 2 w T (t)w(t) \dt By the inequality in (20), then we get f f x T (t)Qx(t)dt <E[V(x(0))] + p 2 E \ tf w T {t)w{t)dt This is the inequality in (9). If x(0) = , then we get the inequality in (8). (A10) 7.3 Appendix C: Proof of Lemma 1 For \e x e 2 e 3 e 4 e 5 e 6 ] ^ , if (25)-(26) hold, then r -iT r e i e 2 < e± . e 6_ a 23 «36 u 21 ^32 b 42 ^33 u 24 b u 0] w e 2 > % e 4 e 5 0J %_ a n a 41 ^22 a 14 a ^23 u 32 W33 14 w 15 a u 55 «36 tu w e 2 V T e 3 + ^2 ^4 % e 5 _^5_ U>_ ^11 hi fc. ^23 '32 ^33 '42 " V b 24 ^2 e 3 ^44 _ %_ <0 This implies that (27) holds. Therefore, the proof is completed. Robust H M Tracking Control of Stochastic Innate Immune System Under Noises 113 7.4 Appendix D: Parameters of the Fuzzy System, control gains and observer gains The nonlinear innate immune system in (33) could be approximated by a Takagi-Sugeno Fuzzy system. By the fuzzy modeling method (Takagi & Sugeno, 1985), the matrices of the local linear system A z , the parameters B , C , D , K and L z are calculated as follows: 3 -1 -0.5 1 -1.5 0.5 -1 -2 9 -1 -1.5 1 -1.5 0.5 -1 -4 15 -1 -2.5 1 -1.5 0.5 -1 . Ao 3 -1 -0.5 1 -1.5 0.5 -1 -2 0" -9 -1 -1.5 1 -1.5 0.5 -1 " -4 -15 -1 -2.5 1 -1.5 0.5 -1 3 -1 -0.5 1 0.5 -2 9 -1 -1.5 1 0.5 " -4 15 -1 -2.5 1 0.5 0" -1 1 -1.5 -1 0" 1.5 -1 -1.5 -1 B = -1 0" -1 1 , c = -1 1 0" 1 , D = 1 1 0" 1 1 1 17.712 0.14477 0.20163 18.201 0.51947 -0.31484 -0.43397 0.37171 -13.967 0.28847 0.0085838 0.046538 0.18604 -0.00052926 -0.052906 14.392 , ; = l,-,9 L,= 12.207 -26.065 22.367 93.156 -8.3701 7.8721 -8.3713 20.912 -16.006 7.8708 -16.005 14.335 i = l, 8. References Asachenkov, A.L. (1994) Disease dynamics. Birkhauser Boston. Balas, G., Chiang, R., Packard, A. & Safonov, M. (2007) MATLAB: Robust Control Toolbox 3 User's Guide. The MathWorks, Inc. Bell, D.J. & Katusiime, F. (1980) A Time-Optimal Drug Displacement Problem, Optimal Control Applications & Methods, 1, 217-225. 114 Robust Control, Theory and Applications Bellman, R. (1983) Mathematical methods in medicine. World Scientific, Singapore. 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Many real-world processes and systems can be modeled as switched systems, such as the automobile direction-reverse systems, computer disk systems, multiple work points control systems of airplane and so on. Therefore, the switched systems have the wide project background and can be widely applied in many domains (Wang, W. & Brockett, R. W., 1997; Tomlin, C. et al., 1998; Varaiya, P., 1993). Besides switching properties, when modeling a engineering system, system uncertainties that occur as a result of using approximate system model for simplicity, data errors for evaluation, changes in environment conditions, etc, also exit naturally in control systems. Therefore, both of switching and uncertainties should be integrated into system model. Recently, study of switched systems mainly focuses on stability and stabilization (Sun, Z. D. & Ge, S. S., 2005; Song, Y. et al, 2008; Zhang, Y. et al, 2007). Based on linear matrix inequality technology, the problem of robust control for the system is investigated in the literature (Pettersson, S. & Lennartson, B., 2002). In order to guarantee Hx> performance of the system, the robust Hx> control is studied using linear matrix inequality method in the literature (Sun, W. A. & Zhao, J., 2005). In many engineering systems, the actuators may be subjected to faults in special environment due to the decline in the component quality or the breakage of working condition which always leads to undesirable performance, even makes system out of control. Therefore, it is of interest to design a control system which can tolerate faults of actuators. In addition, many engineering systems always involve time delay phenomenon, for instance, long-distance transportation systems, hydraulic pressure systems, network control systems and so on. Time delay is frequently a source of instability of feedback systems. Owing to all of these, we shouldn't neglect the influence of time delay and probable actuators faults when designing a practical control system. Up to now, research activities of this field for switched system have been of great interest. Stability analysis of a class of linear switching systems with time delay is presented in the literature (Kim, S. et al., 2006). Robust Hoc control for discrete switched systems with time-varying delay is discussed 118 Robust Control, Theory and Applications in the literature (Song, Z. Y. et al., 2007). Reliable guaranteed-cost control for a class of uncertain switched linear systems with time delay is investigated in the literature (Wang, R. et al., 2006). Considering that the nonlinear disturbance could not be avoided in several applications, robust reliable control for uncertain switched nonlinear systems with time delay is studied in the literature (Xiang, Z. R. & Wang, R. H., 2008). Furthermore, Xiang and Wang (Xiang, Z. R. & Wang, R. H., 2009) investigated robust Loo reliable control for uncertain nonlinear switched systems with time delay. Above the problems of robust reliable control for uncertain nonlinear switched time delay systems, the time delay is treated as a constant. However, in actual operation, the time delay is usually variable as time. Obviously, the system model couldn't be described appropriately using constant time delay in this case. So the paper focuses on the system with time-varying delay. Besides, it is considered that Hoo performance is always an important index in control system. Therefore, in order to overcome the passive effect of time-varying delay for switched systems and make systems be anti-jamming and fault-tolerant, this paper addresses the robust Hx> reliable control for nonlinear switched time-varying delay systems subjected to uncertainties. The multiple Lyapunov-Krasovskii functional method is used to design the control law. Compared with the multiple Lyapunov function adopted in the literature (Xiang, Z. R. & Wang, R. H., 2008; Xiang, Z. R. & Wang, R. H., 2009), the multiple Lyapunov-Krasovskii functional method has less conservation because the more system state information is contained in the functional. Moreover, the controller parameters can be easily obtained using the constructed functional. The organization of this paper is as follows. At first, the concept of robust reliable controller, y -suboptimal robust Hoo reliable controller and y -optimal robust Hoo reliable controller are presented. Secondly, fault model of actuator for system is put forward. Multiple Lyapunov- Krasovskii functional method and linear matrix inequality technique are adopted to design robust Hoo reliable controller. Meanwhile, the corresponding switching law is proposed to guarantee the stability of the system. By using the key technical lemma, the design problems of y -suboptimal robust Hoo reliable controller and y -optimal robust Hoo reliable controller can be transformed to the problem of solving a set of the matrix inequalities. It is worth to point that the matrix inequalities in the y -optimal problem are not linear, then we make use of variable substitute method to acquire the controller gain matrices and y -optimal problem can be transferred to solve the minimal upper bound of the scalar y . Furthermore, the iteration solving process of optimal disturbance attenuation performance y is presented. Finally, a numerical example shows the effectiveness of the proposed method. The result illustrates that the designed controller can stabilize the original system and make it be of Hoo disturbance attenuation performance when the system has uncertain parameters and actuator faults. Notation Throughout this paper, A T denotes transpose of matrix A, L 2 [0,oo) denotes space of square integrable functions on [0,oo) . \\x(t)\\ denotes the Euclidean norm. I is an identity matrix with appropriate dimension. diag{a i } denotes diagonal matrix with the diagonal elements a { , i = l,2,'--,q. S<0 (or S>0) denotes S is a negative (or positive) definite symmetric matrix. The set of positive integers is represented by Z + . A < B (or A > B ) denotes A-B is a negative (or positive) semi-definite symmetric matrix. * in A B] T represents the symmetric form of matrix, i.e. * = B . * C Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 119 2. Problem formulation and preliminaries Consider the following uncertain switched nonlinear system with time-varying delay m = K(,f(t) + A drj(t) x(t - d(t)) + B a{t)U f (t) + D a(t) w{t) + f a(t) (x(t),t) (1) z(t) = C a(t) x{t) + G a(t)U f{t) + N„ (t) w(t) (2) x(f) = #),fe[-p,0] (3) where x(t) e R m is the state vector, w(t) e R q is the measurement noise, which belongs to L 2 [0,oo) , z(t) e R v is the output to be regulated, w (t) eR is the control input of actuator fault. The function a(t) :[0,oo) -^ N = {1,2,- --,1V} is switching signal which is deterministic, piecewise constant, and right continuous, i.e. a(t) :{(0,cr(0)),(^ 1 , <j(t x )),•••, (t k ,cr(t k ))},k eZ + , where t k denotes the k th switching instant. Moreover, a(t) = i means that the i th subsystem is activated, N is the number of subsystems. </>(t) is a continuous vector-valued initial function. The function d(t) denotes the time-varying state delay satisfying < d(t)< p <oo,d(t)< ju <1 for constants p , ju , and /j-(v): R m xR^R m for ieN are unknown nonlinear functions satisfying ||/;(x(o,f)||^IMO|| (4) where U i are known real constant matrices. The matrices A i , A di and B { for i e N are uncertain real-valued matrices of appropriate dimensions. The matrices A i , A di and B i can be assumed to have the form [A i ,A, ! ,B ! ] = [A ! ^*^ ! .] + H j F j (t)[E li ,E, i ,E 2! ] (5) where A z ,A^,B z ,H z ,E lz ,E^ and E 2i for ieN are known real constant matrices with proper dimensions, H z ,E lz ,E^ and E 2i denote the structure of the uncertainties, and F^t) are unknown time-varying matrices that satisfy Fim(t)<i (6) The parameter uncertainty structure in equation (5) has been widely used and can represent parameter uncertainty in many physical cases (Xiang, Z. R. & Wang, R. H., 2009; Cao, Y. et al, 1998). In actual control system, there inevitably occurs fault in the operation process of actuators. Therefore, the input control signal of actuator fault is abnormal. We use u(t) and w (t) to represent the normal control input and the abnormal control input, respectively. Thus, the control input of actuator fault can be described as uf(t) = M,u(t) (7) where M t is the actuator fault matrix of the form M i =diag{m il ,m i2 ,~-,m il ], < m ik < m ik < m ik , m ik >l, k = l,2,~',l (8) 120 Robust Control, Theory and Applications For simplicity, we introduce the following notation M i0 =diag{m il ,m i2 ,'--,fh il } (9) Ji= di "g{jil>ji2>~',ju} ( 10 ) L i =diag{l il ,l i2 ,.-,l il } (11) i 1/- x • m ik~ m ik i m ik~ where m ik = -{m ik + m ik ) , ] ik = _ lK ~ lK , l ik = -^- 2\ IK — IK/ ' J IK — 'IK ~ m ik + mik m ik By equation (9)-(ll), we have M,=M, (I + L,),|L,|<J,<7 (12) where |L f | represents the absolute value of diagonal elements in matrix L z , i.e. M=^{|z n |,y,-,y Remark 1 m zfc = 1 means normal operation of the k th actuator control signal of the i th subsystem. When m ik = , it covers the case of the complete fault of the k th actuator control signal of the i th subsystem. When m ik > and m ik ^ 1 , it corresponds to the case of partial fault of the k th actuator control signal of the i th subsystem. Now, we give the definition of robust fix, reliable controller for the uncertain switched nonlinear systems with time- varying delay. Definition 1 Consider system (1) with w(t ) = . If there exists the state feedback controller u(t) = K a ^x(t) such that the closed loop system is asymptotically stable for admissible parameter uncertainties and actuator fault under the switching law <j(t) , u(t) = K a ^x(t) is said to be a robust reliable controller. Definition 2 Consider system (l)-(3). Let />0 be a positive constant, if there exists the state feedback controller u(t) = K a ^x(t) and the switching law a(t) such that i. With w(t ) = , the closed system is asymptotically stable. ii. Under zero initial conditions, i.e. x(t) = (t e [-/?,0]) , the following inequality holds |z(0| 2 < r\H)\\ 2 > V M0 g l 2 [o /Q o) , w(t) * o (13) u(t ) = K a ^x(t ) is said to be y -suboptimal robust Hoc reliable controller with disturbance attenuation performance y . If there exists a minimal value of disturbance attenuation performance y , u(t) = K a ^x(t) is said to be y -optimal robust H*, reliable controller. The following lemmas will be used to design robust Hoc reliable controller for the uncertain switched nonlinear system with time-varying delay. Lemma 1 (Boyd, S. P. et al., 1994; Schur complement) For a given matrix s = S n S 12 S 21 S 22 with S n = S^ , S 22 = S \ 2 , S 12 = S \\ , then the following conditions are equivalent i S<0 II iZ?-i i \ u i '-'22 21 1 1 1 2 III ^22 / 11 12 22 21 Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 121 Lemma 2 (Cong, S. et al., 2007) For matrices X and Y of appropriate dimension and Q > , we have X T Y + Y T X < X T QX + Y T Q~ 1 Y Lemma 3 (Lien, C.H., 2007) Let Y,D,E and F be real matrices of appropriate dimensions with F satisfying F T =F , then for all F T F < I Y + DFE + E T F T D T < if and only if there exists a scalar s > such that Y + eDD t + s~ x F?F < Lemma 4 (Xiang, Z. R. & Wang, R. H., 2008) For matrices R lf R 2 , the following inequality holds R 1 Z(t)R 1 + R T 2 I T {t)Rl < pR x URl + p- 1 R\UR 1 where j3 > , Z(i) is time-varying diagonal matrix, U is known real constant matrix satisfying \Z(t)\ < U , \£(t)\ represents the absolute value of diagonal elements in matrix S{t). Lemma 5 (Peleties, P. & DeCarlo, R. A., 1991) Consider the following system m=U)(m (i4) where cr(t) :[0,oo) -» N = {1,2,- --,N} . If there exist a set of functions V t :R m -> R, z e N such that (i) V z is a positive definite function, decreasing and radially unbounded; (ii) &V { (x(t))/dt = (dV i /dx)fi (x)<0 is negative definite along the solution of (14); (iii) Vj(x(t k ))<Vi(x(t k )) when the i th subsystem is switched to the ; th subsystem i,jsN, i * j at the switching instant t k , k = Z + , then system (14) is asymptotically stable. 3. Main results 3.1 Condition of stability Consider the following unperturbed switched nonlinear system with time-varying delay x(t) = A a{t) x{t) + A Mt) x{t - d(t)) + f a(t) (x(t), t) (15) x(t) = #t),tel-p,0] (16) The following theorem presents a sufficient condition of stability for system (15) -(16). Theorem 1 For system (15)-(16), if there exists symmetric positive definite matrices F U Q, and the positive scalar 8 such that F { < SI (17) 122 Robust Control, Theory and Applications AjPi + P^+R+Q + SUfUi P,A dl <0 (18) -(I-A)QJ where i * j, i,j e N , then systems (15)-(16) is asymptotically stable under the switching law o(t) = argmin{x T (f)f;x(f)} • z'eN Proof For the i th subsystem, we define Lyapunov-Krasovskii functional V,(x(t)) = X T (t)PMt) + H m X T (T)Qx(T)dT where P { ,Q are symmetric positive definite matrices. Along the trajectories of system (15), the time derivative of V t (t) is given by V,(x(t)) = x T (t)P lX (t) + x T (t)P,x(t) + x T (t)Qx(t) - (1 - d(t))x T (t - d(t))Qx(t - d(t)) < x T (t)P,x(t) + x r {t)P i x{t) + x T (t)Qx(t) - (1 - ju)x T (t - d(t))Qx(t - d(t)) = 2x T (t)P,[A,x(t) + A di x(t - d(t)) + Mx(t),t)] + x T (t)Qx(t) -(l-ju)x T (t-d(t))Qx(t-d(t)) = x T (t)(AjP, + P,A, +Q)x(t) + 2x T (t)P,A d ,x(t - d(t)) + 2* T (t)P,/,(*(t),') -(l-ju)x T (t-d(t))Qx(t-d(t)) From Lemma 2, it is established that ix T (t )p i / i ( X (t), t) < x 1 {mm + f! w), t)pj { ( X (t), t) From expressions (4) and (17), it follows that ixTitWMW^x^PMn+sfhmMimV^xTim+mJuMt) Therefore, we can obtain that V, (x(t)) < x T (t)(A[P, + P,A, +Q + P.+ SUjU,)x(t) + 2x T (t)P,A dl x(t - d(t)) -(l-ju)x T (t-d(t))Qx(t-d(t)) where rj = x(t) x(t-d(t)) From (18), we have 6>,= AjP i+ PA +Q + P i +SUjU i P t A dl -(1-/0(2. i +diag{P j -P i ,G\<O (19) Using rj and rj to pre- and post- multiply the left-hand term of expression (19) yields v 1 (x(t))<x T (t)(P,-P j Mt) (20) Robust H„ Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 123 The switching law a(t) = argmin{;t r (f)P ; x(f)} expresses that for i,j eN,i* j , there holds the inequality " (20) and (21) lead to x T (t)P,x(t)<x T (t)P jX (t) W))<0 (21) (22) Obviously, the switching law <j(t) = argmin{x T (t)Px(t)} also guarantees that Lyapunov- Krasovskii functional value of the activated subsystem is minimum at the switching instant. From Lemma 5, we can obtain that system (15)-(16) is asymptotically stable. The proof is completed. ■ Remark 2 It is worth to point that the condition (21) doesn't imply P { < P , for the state x(t) doesn't represent all the state in domain R m but only the state of the i th activated subsystem. 3.2 Design of robust reliable controller Consider system (1) with w(t) = m = K ( ,f(t) + A d!T(t) x(t - d(t)) + K (t) u f {t) + L ( ,)(mt) x(t) = tft),te[-p,0] By (7), for the i th subsystem the feedback control law can be designed as uf(t) = M i K i x(t) Substituting (25) to (23), the corresponding closed-loop system can be written as x(t) = A { x(t) + A di x(t - d(t)) + f { (x(t), t) where A { = A { + ^M^ , i e N . The following theorem presents a sufficient existing condition of the robust reliable controller for system (23) -(24). Theorem 2 For system (23)-(24), if there exists symmetric positive definite matrices X if S , matrix Y i and the positive scalar X such that (23) (24) (25) (26) X, > XI (27) f, Ass Hi fj x* x> x { uj * -{1-H)S SEl * * -I * * * -I * * * * -s * * * * * " X ; * * * * * * -XI <0 (28) 124 Robust Control, Theory and Applications where i * j, i, j e N , l F i = A { X { + B^^ + ( A Z X. + B^^) 1 , ® { = E lz X z + E li bA l Y i , then there exists the robust reliable state feedback controller u(t) = K a(t) x(t),K,=Y,X- 1 (29) and the switching law is designed as a(t) = argmin{x T (£)X z 1 x(t)} , the closed-loop system is asymptotically stable. Proof From (5) and Theorem 1, we can obtain the sufficient condition of asymptotically stability for system (26) R<SI A- P^ + Hfi^E^ <i-m)Q <0 (30) (31) and the switching law is designed as <j(t) = argmin{x (t)Pix(t)} , where 4. = P,[A, + B t M t K t + H,F,(f)(E 1( + E 2 ,M,K,)] + [A + B t M t K t + H t F t (t)(E„ + E 2t M t K t )] T P t + P j+ Q + SUjU, Denote Y„ P { (A, + BiH-K,- ) + (A + B.M.K, ) T P l + P ]+ Q + SUjU, P { A di (32) Then (31) can be written as Xw ^•W[E li +E 2i M i X i E^ + [E lz+ E 2z M^ £*]'*?(*) ^ < (33) By Lemma 3, if there exists a scalar £ > such that Y lj+ s mi -^- 1 [E li + E 2i M^ E, z ] i [E lz+ E 2z M^. E dl ]<0 (34) then (31) holds. (34) can also be written as where Hij PiAx+e-^Eu+EuMfcfE* -(l- M )Q + £ -%, T E dl <0 (35) Il^^Ai+BMiKifPi+PiiAi+BMiKd + s^HiHjPi+s-^+E^l^Kf^+EiiMiKi) + P l +Q + SU T i U i Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 125 Vi cYix Using diag{s /2 ,s /2 } to pre- and post- multiply the left-hand term of expression (35) and denoting P { = eP i ,Q = eQ , we have n tj P i A ii+ {E li+ E 2i M i K i ) T E di -(l- M )Q + E dt T E dt <0 (36) where n„ = (A, + B^KfP, + P^A, + B..M..K,.) + P,H,HjP, + (E li + E 2l M,K,) T (E v + E 2 ,.M,.K,.) By Lemma 1, (36) is equivalent to n (j p ( A di v x n x (E li + E 2i M f x f ; * -(1-//)Q E T di -I <0 (37) where ri {] = (A i + B^Mfr f P i + P { (A i + B^M^ ) + Pj+Q + sSUjU t Using diag{Pj~ ,Q _1 ,1,1} to pre- and post- multiply the left-hand term of expression (37) and denoting X { = Pr 1 ,Y X = K^ 1 ,S = Q~ 1 ,X = (sS)' 1 , (37) can be written as ~ n l A it S Hi (Ei x, + E 2t M t Y t f * -(l-A)S SEl * * -I ■k * * -I < (38) where I7-j = (A i X i + B^ftf + (A { + B i M i Y i ) + X t {Xf + S" 1 + X^UjU^ Using Lemma 1 again, (38) is equivalent to (28). Meanwhile, substituting X z = P[ ,P { - sP { and X = (sSy 1 to (30) yields (27). Then the switching law becomes cr(t) = aigmin{x T (t)X^x(t)} (39) Based on the above proof line, we know that if (27) and (28) holds, and the switching law is designed as (39), the state feedback controller u(t) = K a , t sx(t) , K x = YjX^ 1 can guarantee system (23) -(24) is asymptotically stable. The proof is completed. ■ 3.3 Design of robust H*, reliable controller Consider system (l)-(3). By (7), for the i th subsystem the feedback control law can be designed as 126 Robust Control, Theory and Applications u f (t) = M i K i x(t) (40) Substituting (40) to (1) and (2), the corresponding closed-loop system can be written as (41) (42) x(t) = A { x(t) + A dl x(t - d(t)) + D { w(t) + /;• (x(t), t) z(t) = C i x(t) + N i w(t) where A f = A { + B^K^q = C { + QM^ , i e N . The following theorem presents a sufficient existing condition of the robust Hoo reliable controller for system (l)-(3). Theorem 3 For system (l)-(3), if there exists symmetric positive definite matrices X-,S, matrix Y t and the positive scalar X,s such that X; > XI (43) '5P5 D t - (C I -X I .+G t .M I .Y i ) T -ysl N A d ,S H z 0/ x,- X; x t .u/ (1-/0S s4 * -I * * -I * * * -s * * * * " X ; * * * * * -XI <0 (44) where i * j, i,jeN, x F i = A Z X Z + B^^ + (A Z X Z + B f M f Y;.) J , & t = E lz X z + E^M^ , then there exists the robust Hoo reliable state feedback controller u{t) = K a(t) x{t),K,=Y,Xf (45) and the switching law is designed as <j(t) = argmin{x T (^)X i ~ 1 x(^)} / the closed-loop system is z'eN asymptotically stable with disturbance attenuation performance y for all admissible uncertainties as well as all actuator faults. Proof By (44), we can obtain that W { A d ,S H, <Z> T * -(l-ju)S SE T d , * * -I x> x, x t ut -s * - x ; * * -XI <0 (46) Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 127 From Theorem 2, we know that closed-loop system (41) is asymptotically stable. Define the following piecewise Lyapunov-Krasovskii functional candidate V{x(t)) = V l (x{t)) = x T {t)P l x{t) + \l d(t) x T (T)Qx{T)dr, te[t n , t n+1 ), n = 0,l,- (47) where P z , Q are symmetric positive definite matrices, and t = . Along the trajectories of system (41), the time derivative of V { (x(t)) is given by Vi(x(t))<{ 1 4 P t D t P,(A d , + H,F,(t)E dt ) -0--m)Q * * (48) where Z = [x T (t) w T (t) x T (t-d(t))J , 4 = P^[A { + B^Kt + H^(0(E lz + E2..M..JK,.)] + [A { + B Z M Z K Z + H^(0(E lz + E 2z M^)] T ^ + P i +Q + SUjU i By simple computing, we can obtain that y- 1 z T (t)z(t)-yw T (t)w(t) y-\C x + G Z M^) T (Q + G f M f ig ^(Q + QM,^ ) T N, o" * * o (49) Denote X z = If \Y; =K z If \S = Q~\l^ =£^,Q = ^Q . Substituting them to (44), and using Lemma 1 and Lemma 3, through equivalent transform we have y-\C, + G ) .M,.K f ) T (C ) . + G..M..K,.) + 4 ^(C, + G.-M^,-) 1 ^ + ^D, P,.^ + H,F,(f)£ d ,) y^NfNi-yl -(1-//X2 where <0 (50) 4, = ^[A ; +5^^ +H i F i (t)(E li +E 2i M^ i )] + lA + B i M i K i +H i F i (t)(E li +£ 2! M i K i )] T P i + P ; +Q + <yLZ, T U,. Obviously, under the switching law cx(r) = argmin{x r (r).F;x(r)} there is r _ V(0z(f) - rw T (t)w(f) + v;(x(0) < (51) Define } = ^(y- 1 z T (t)z(t)-yw T (t)w(t))dt (52) 128 Robust Control, Theory and Applications Consider switching signal ;(°h ;M < 2 h #h which means the v ' th subsystem is activated at t k . Combining (47), (51) and (52), for zero initial conditions, we have / < f {j- x z\t)z(t) - 7 w T (t)w(t) + y, 0) (t))dt + f (^V (0z(0 - r ^ T (0^(0 + v. (1) (t))d* + • • • <o Therefore, we can obtain \\z(t)\\ 2 < y\\w(t)\\ 2 . The proof is completed. ■ When the actuator fault is taken into account in the controller design, we have the following theorem. Theorem 4 For system (l)-(3), y is a given positive scalar, if there exists symmetric positive definite matrices X Z ,S, matrix Y i and the positive scalar a,s,X such that X, > XI (53) -ysl Nj where i * j,i,j sN A dl S H t x* *i x t u; ViMioh aGJfil (1-//)S SE T di * -I * * ** * * * -s * * * * " X ; * * * * * -XL * * * * * * -al <0 (54) I, = A,X, + B,M, o y, + (A,X, + B,M, V;-) r + aBjtf ir 1( =c,x,4 I 2i = E lj X i ■ ^E 2t M, Y,+aE 2 J t Bj £ -L + aE 2i J { E 2i then there exists the y -suboptimal robust H^ reliable controller u(t) = K (7{t) x(t),K,=Y,X- 1 (55) Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 129 and the switching law is designed as a(t) = argmin{x T (£)X z 1 x(t)} , the closed-loop system is asymptotically stable. Proof By Theorem 3, substituting (12) to (44) yields B,-AWi- (B,M, L,.Y ; ) (g^l^y (E 2l M l0 * * * * * * * * * * o" * * * * * * :0 ( 56 ) where W i0 D i (C,X,.+G,M, Y ; ) -ysl -U iPSo = A,X, + B,M l0 Y, + (A,X, + B,M i0 Y,) <H =E Ji Denote 0, o =E v X 1+ E 2l M, o Y, US Hi <9o *i *i x t u{ -fi)S SEl * -I * * -I * * * -s * * * * " X ; * * * * * -XI B,.M, L,.y,+(B,.M, L,.Y,.) (G^-oI,^ (EnM i0 •k * •k * * * * * * * o~ * * * * * * (57) 130 Robust Control, Theory and Applications Notice that M i0 and L z are both diagonal matrices, then we have L^M^ 0] + G { En From Lemma 4 and (12), we can obtain that L z [M z0 Y z 0000000 0] E:<a ~V "V T y z t m z0 " ~Y?M i0 ~ G { Q Ji + a~ 1 Ji E 2i E 2i Then the following inequality (58) ~B;~ ~B;~ T X T M i0 " ~Y?M i0 ~ Gi Gi Ji + CT 1 Ji E 2i E 2i <0 (59) can guarantee (56) holds. By Lemma 1, we know that (59) is equivalent to (54). The proof is completed. ■ Remark 3 (54) is not linear, because there exist unknown variables s , s~ x . Therefore, we consider utilizing variable substitute method to solve matrix inequality (54). Using diag{1 , 8~ x ,1 ,1 ,1 ,1 ,1 ,1 ,1 ,1} to pre- and post- multiply the left-hand term of expression (54), and denoting rj = e~ x , (54) can be transformed as the following linear matrix inequality Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 131 3 rpi A di S ^ ^2i X; *i x { uj Y?M i0 J% * -rjyl inJ * k z* aGJfili * k * -(1- M )S SE T di * k * k -I * k * k k Z±i k k k k k k -s k k k k k k * " X ; k k k k k k * * -XI k k k k k k * * k -al <0 (60) where 27 3i = -rjyl + aG i ] i G i Corollary 1 For system (l)-(3), if the following optimal problem mm y X- >0, S>0, a>0, s>0, A>0,Y { (61) s.t. (53) and (54) has feasible solution X { > 0,S > 0,a > 0,s > 0,X > 0,Y if i e N , then there exists the y -optimal robust Hoc reliable controller u(t) = K a(t) x(t),K,=Y,X- 1 (62) and the switching law is designed as a(t) = argmin{x T (£)X z 1 x(t)} , the closed-loop system is asymptotically stable. Remark 4 There exist unknown variables ye , ys~ x in (54), so it is difficult to solve the above optimal problem. We denote 6 = ys , % = ys~ x , and substitute them to (54), then (54) becomes a set of linear matrix inequalities. Notice that y < %. , we can solve the following 2 the optimal problem to obtain the minimal upper bound of y s.t. Xi>0,S> mm O,a>O,0>O,z>O,A >0,Yi 2 I { D t A d ,S H { z 2i x. x. X i U I Y?M i0J k -61 Nj k k Zn aGJfil k k k -(1-M)S SE T di k k k k -I k k k k k *u k k k k k k -s k k k k k k k " X ; k k k k k k k * -XI k k k k k k k * k -al <0 (63) (64) 132 Robust Control, Theory and Applications X, > XL (65) where Z 3i = -%L + aG i ] i G i , then the minimal value of y can be acquired based on the following steps Step 1. From (63)-(65), we solve the minimal value y^ of — , where y^ is the first iterative value; Step 2. Choosing an appropriate step size 8 = S , and let y^ -y^ -S , then we substitute Step 3. y^ 1 ' to (60) to solve LMIs. If there is not feasible solution, stop iterating and y^ ' is just the optimal performance index; Otherwise, continue iterating until y^ ' is feasible solution but y^ +1 ' is not, then y = y( ' - kS is the optimal performance index. 4. Numerical example In this section, an example is given to illustrate the effectiveness of the proposed method. Consider system (l)-(3) with parameter as follows (the number of subsystems N = 2) ~-2 0" 2 , A 2 = ~1 -3" -2 / A*i = "-5 - 0" -4 > A d2 = "-3 -1" -6 , B 1 = "-5 7" -9 ,B 2 = "-8 2" 2 6 "2 5" , E 12 = ~1 2" 4 , E 21 = "2 0" 3 1 , E 22 = " 2 0" 0.2 / E ^i = -1 " 1 0.1 ' ^dl = "2 0" 1 C a =[-0.8 0.5],C 2 =[0.3 -0.8] , G a = [0.1 0],G 2 =[-0.1 0],D a : 2 -1 -4 D 7 3 -6 -5 12 , H 1 =H 2 0.1 0.2 0.1 , N 1 =N 2 = [0.01 0] The time-varying delay d(t) = 0.5e f , the initial condition x(t) = [2 -l] , t e [-0.5,0] , Tsinf uncertain parameter matrices F a (f) = F 2 (f) = sinf , / = 0.8 and nonlinear functions is selected as fMtyt)-- x a (f)cos(f) f 2 (x(t),t): x 2 (t)cos(t) then U a = 1 ,U 2 = 1 When M 1 = M 2 = I , from Theorem 3 and using LMI toolbox in Matlab, we have X 1 0.6208 0.0909 0.0909 0.1061 X 9 0.2504 0.1142 0.1142 0.9561 *i = 0.6863 0.5839 -3.2062 -0.3088 0.8584 -1.3442 -0.5699 -5.5768 Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 133 '0.1123 0.0370" 0.0370 0.1072 , s = 19.5408, 1 = 0.0719 Then robust Hao controller can be designed as 0.3426 5.2108' -5.4176 1.7297 ,K 2 4.3032 -1.9197" 0.4056 -5.8812 Choosing the switching law a(t) = argmin{x T (f)X z 1 x(t)} , the switching domain is z'eN f\ = {x(t) e R 2 \x T (t)X^x(t) < x T (t)X^x(t)} 0, = {x(t) e R 2 \x T (t)X^x(t) > x T {t)X^x{t)} O l ={x(t)eR 2 \x T (t) n l ={x(t)eR 2 \x T (t) The switching law is a(t)- -2.3815 -1.0734 -1.0734 9.6717 -2.3815 -1.0734" -1.0734 9.6717 1 x(t) g r\ x(t) < 0} x(t) > 0} [2 x(t)eQ The state responses of the closed-loop system are shown in Fig. 1. 1 L 1 1 L 1 J L 1 J 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x1 I i i * \ \ \ \ \ \ \ \ \ h 1 I h I \ \ I \ I ! ! ! ! ! ! ! ! /\ i y i i i i i i / ! ! ! ! ! ! ! ! k — \ — r 1 i r 1 i x 1 1 x2 o o.2 0.4 o.e 1 1.2 1.4 1.( t/s Fig. 1. State responses of the closed-loop system with the normal switched controller when the actuator is normal 134 Robust Control, Theory and Applications The Fig. 1 illustrates that the designed normal switched controller can guarantee system is asymptotically stable when the actuator is normal. However, in fact, the actuator fault can not be avoided. Here, we assume that the actuator fault model with parameters as follows For subsystem 1 For subsystem 2 Then we have 0.04<ra n <l, 0.1<m 12 <1.2 0.1 < m 21 < 1 , 0.04 < m 22 < 1 M 10 ' M 20 = 0.52 0.65 0.55 0.52 h h = 0.92 0.85 0.82 0.92 Choosing the fault matrices of subsystem 1 and subsystem 2 are M, 0.04 " 0.1 , M 2 = 0.1 0.04 Then the above switched controller still be used to stabilize the system, the simulation result of the state responses of closed-loop switched system is shown in Fig. 2. ra X1I I I I I I I I \y i 1- 1 1 i ^ \ 1 i i x2 i i i i i i i A I I I I I I II ^ ' T 1 1 I I 1 1 I I ! ! ! ! ! ! ! ! if ; ; ; v " ; ; ; "\ ; ft f ~ T ~ ~ 1 — ! ~ ~ " — ! " " "1 — ! — ! — [~ ~ " \\ ! ! ! ! ! ! ! ! 10 1 t/s 2 14 16 18 20 Fig. 2. State responses of the closed-loop system with the normal switched controller when the actuator is failed Obviously, it can be seen that system state occurs vibration and the system can not be stabilized effectively. The simulation comparisons of Fig. 1 and Fig. 2 shows that the design method for normal switched controller may lose efficacy when the actuator is failed. Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 135 Then for the above fault model, by Theorem 3 and using LMI toolbox in Matlab, we have X, Yi = 0.0180 0.0085 0.0085 0.0123 0.4784 0.6606 " -0.5231 -0.0119 X 9 0.0436 -0.0007 -0.0007 0.0045 "0.7036 -0.1808' -0.1737 -0.5212 0.0130 0.0000 0.0000 0.0012 , a = 0.0416, * = 946.1561, 1 = 0.0036 Then robust H x reliable controller can be designed as 1.8533 52.3540' -42.5190 28.3767 , K 9 15.5242 -37.5339 -5.8387 -116.0295 Choosing the switching law as a(t)- 1 x(t)^£\ 2 x(t)eQ n l ={x(t)eR 2 \x T (t) 1 ={x(t)eR 2 \x T (t) where ., m T 59.6095 -60.5390" x(t)<0} x(t)>0} The state responses of the closed-loop system are shown in Fig. 3. 59.6095 -60.5390 -60.5390 -100.9210 59.6095 -60.5390 -60.5390 -100.9210 0-/-V- A - - -^^: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.< t/s Fig. 3. State responses of the closed-loop system with the reliable switched controller when the actuator is failed 136 Robust Control, Theory and Applications It can be seen that the designed robust Hoo reliable controller makes the closed-loop switched system is asymptotically stable for admissible uncertain parameter and actuator fault. The simulation of Fig. 3 also shows that the design method of robust Hoo reliable controller can overcome the effect of time-varying delay for switched system. Moreover, by Corollary 1, based on the solving process of Remark 4 we can obtain the optimal Hoo disturbance attenuation performance y = 0.54 , the optimal robust Hoo reliable controller can be designed as K 1 = 9.7714 115.4893 -69.8769 41.1641 ,K 2 = 9.9212 -62.1507 -106.5624 -608.0198 The parameter matrices X 1 , X 2 of the switching law are X-i = , X 2 0.0031 0.0011 0.0011 0.0018 0.0119 -0.0011 -0.0011 0.0004 5. Conclusion In order to overcome the passive effect of time-varying delay for switched systems and make systems be anti-jamming and fault-tolerant, robust Hoo reliable control for a class of uncertain switched systems with actuator faults and time- varying delays is investigated. At first, the concept of robust reliable controller, y -suboptimal robust Hoo reliable controller and y -optimal robust Hoo reliable controller are presented. Secondly, fault model of actuator for switched systems is put forward. Multiple Lyapunov-Krasovskii functional method and linear matrix inequality technique are adopted to design robust Hoo reliable controller. The matrix inequalities in the y -optimal problem are not linear, then we make use of variable substitute method to acquire the controller gain matrices. Furthermore, the iteration solving process of optimal disturbance attenuation performance y is presented. Finally, a numerical example shows the effectiveness of the proposed method. The result illustrates that the designed controller can stabilize the original system and makes it be of Hoo disturbance attenuation performance when the system has uncertain parameters and actuator faults. Our future work will focus on constructing the appropriate multiply Lyapunov-Krasovskii functional to obtain the designed method of time delay dependent robust Hoo reliable controller. 6. Acknowledgment The authors are very grateful to the reviewers and to the editors for their helpful comments and suggestions on this paper. This work was supported by the Natural Science Foundation of China under Grant No. 60974027. 7. References Boyd, S. P.; Ghaoui, L. E. & Feron, et al. (1994). Linear matrix inequalities in system and control theory. SIAM. Robust H M Reliable Control of Uncertain Switched Nonlinear Systems with Time-varying Delay 137 Cao, Y.; Sun, Y. & Cheng, C. (1998). Delay dependent robust stabilization of uncertain systems with multiple state delays. 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IEEE Transactions on Automatic Control, Vol. 38, No. 2, 195-207. Wang, W. & Brockett, R. W. (1997). Systems with finite communication bandwidth constraints-part I: State estimation problems. IEEE Transactions on Automatic Control, Vol. 42, No. 9, 1294-1299. Wang, R.; Liu, J. C. & Zhao, J. (2006). Reliable guaranteed-cost control for a class of uncertain switched linear systems with time-delay. Control Theory and Applications, Vol. 23, No. 6, 1001-1004. (in Chinese) Xiang, Z. R. & Wang, R. H. (2008). Robust reliable control for uncertain switched nonlinear systems with time delay. Proceedings of the 7th World Congress on Intelligent Control and Automation, pp. 5487-5491. Xiang, Z. R. & Wang, R. H. (2009). Robust Loo reliable control for uncertain nonlinear switched systems with time delay. Applied Mathematics and Computation, Vol. 210, No. 1, 202-210. 138 Robust Control, Theory and Applications Zhang, Y.; Liu, X. Z. & Shen, X. M. (2007). Stability of switched systems with time delay. Nonlinear Analysis: Hybrid Systems, Vol. 1, No. 1, 44-58. Part 3 Sliding Mode Control Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization Hai-Ping Pang and Qing Yang Qingdao University of Science and Technology China 1. Introduction Optimal control is one of the most important branches in modern control theory, and linear quadratic regulator (LQR) has been well used and developed in linear control systems. However, there would be several problems in employing LQR to uncertain nonlinear systems. The optimal LQR problem for nonlinear systems often leads to solving a nonlinear two-point boundary-value (TPBV) problem (Tang et al. 2008; Pang et al. 2009) and an analytical solution generally does not exist except some simplest cases (Tang & Gao, 2005). Additionally, the optimal controller design is usually based on the precise mathematical models. While if the controlled system is subject to some uncertainties, such as parameter variations, unmodeled dynamics and external disturbances, the performance criterion which is optimized based on the nominal system would deviate from the optimal value, even the system becomes unstable (Gao & Hung, 1993 ; Pang & Wang, 2009). The main control strategies to deal with the optimal control problems of nonlinear systems are as follows. First, obtain approximate solution of optimal control problems by iteration or recursion, such as successive approximate approach (Tang, 2005), SDRE (Shamma & Cloutier, 2001), ASRE (Cimen & Banks, 2004). These methods could have direct results but usually complex and difficult to be realized. Second, transform the nonlinear system into a linear one by the approximate linearization (i.e. Jacobian linearization), then optimal control can be realized easily based on the transformed system. But the main problem of this method is that the transformation is only applicable to those systems with less nonlinearity and operating in a very small neighborhood of equilibrium points. Third, transform the nonlinear system into a linear one by the exact linearization technique (Mokhtari et al. 2006; Pang & Chen, 2009). This differs entirely from approximate linearization in that the approximate linearization is often done simply by neglecting any term of order higher than 1 in the dynamics while exact linearization is achieved by exact state transformations and feedback. As a precise and robust algorithm, the sliding mode control (SMC) (Yang & Ozgiiner, 1997; Choi et al. 1993; Choi et al. 1994) has attracted a great deal of attention to the uncertain nonlinear system control problems. Its outstanding advantage is that the sliding motion exhibits complete robustness to system uncertainties. In this chapter, combining LQR and SMC, the design of global robust optimal sliding mode controller (GROSMC) is concerned. Firstly, the GROSMC is designed for a class of uncertain linear systems. And then, a class of 142 Robust Control, Theory and Applications affine nonlinear systems is considered. The exact linearization technique is adopted to transform the nonlinear system into an equivalent linear one and a GROSMC is designed based on the transformed system. Lastly, the global robust optimal sliding mode tracking controller is studied for a class of uncertain affine nonlinear systems. Simulation results illustrate the effectiveness of the proposed methods. 2. Optimal sliding mode control for uncertain linear system In this section, the problem of robustify LQR for a class of uncertain linear systems is considered. An optimal controller is designed for the nominal system and an integral sliding surface (Lee, 2006; Laghrouche et al. 2007) is constructed. The ideal sliding motion can minimize a given quadratic performance index, and the reaching phase, which is inherent in conventional sliding mode control, is completely eliminated (Basin et al. 2007). Then the sliding mode control law is synthesized to guarantee the reachability of the specified sliding surface. The system dynamics is global robust to uncertainties which satisfy matching conditions. A GROSMC is realized. To verify the effectiveness of the proposed scheme, a robust optimal sliding mode controller is developed for rotor position control of an electrical servo drive system. 2.1 System description and problem formulation Consider an uncertain linear system described by x(t) = (A + AA)x(t) + (B + AB)u(t) + S(x,t) (1) where x(t) e R n and u(t) e R m are the state and the control vectors, respectively. AA and AB are unknown time-varying matrices representing system parameter uncertainties. 8{x,t) is an uncertain extraneous disturbance and/ or unknown nonlinearity of the system. Assumption 1. The pair (A,B) is controllable and rank(B) = m . Assumption 2. AA , AB and 8{x,t) are continuously differentiable inx, and piecewise continuous in t . Assumption 3. There exist unknown continuous functions of appropriate dimension AA , AB and 8(x, t) , such that AA = BAA, AB = BAB, 8{x,t) = BS(x,t). These conditions are the so-called matching conditions. From these assumptions, the state equation of the uncertain dynamic system (1) can be rewritten as x(t ) = Ax(t) + Bu(t) + BS(x, t), (2) where Assumption 4. There exist unknown positive constants y and y 1 such that \S{x f t)\- ; 7o+riR where |«| denotes the Euclidean norm. By setting the uncertainty to zero, we can obtain the dynamic equation of the original system of (1), as Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 143 x(t) = Ax(t) + Bu(t). (3) For the nominal system (3), let's define a quadratic performance index as follows: 7o = |j V( W) + u\t)Ru{t)]dt , (4) where Q e R nxn is a semi-positive definite matrix, the weighting function of states; R e R mxm is a positive definite matrix, the weighting function of control variables. According to optimal control theory and considering Assumptionl, there exists an optimal feedback control law that minimizes the index (4). The optimal control law can be written as u\t) = -R- 1 B T Px(t), (5) where P e R nxn is a positive definite matrix solution of Riccati matrix equation: -PA - A T P + PBR-^P - Q = 0. (6) So the dynamic equation of the closed-loop system is x(t) = (A-BRr 1 B T P)x(t). (7) Obviously, according to optimal control theory, the closed-loop system is asymptotically stable. However, when the system is subjected to uncertainties such as external disturbances and parameter variations, the optimal system behavior could be deteriorated, even unstable. In the next part, we will utilize sliding mode control strategy to robustify the optimal control law. 2.2 Design of optimal sliding mode controller 2.2.1 Design of optimal sliding mode surface Considering the uncertain system (2), we chose the integral sliding surface as follows: s(x, t) = G[x(t) - x(0)] - G J' (A - BR~ 1 B T P)x(T)dT = 0. (8) where G e R mxn t which satisfies that GB is nonsingular, x(0) is the initial state vector. In sliding mode, we have s(x,t) = and s(x,t) = . Differentiating (8) with respect to t and considering (1), we obtain s = G[(A + AA)x + (B + AB)u + S]- G(A - BR~ 1 B T P)x = GAAx + G(B + AB)u + GS + GBR~ 1 B T Px (9) = G(AAx + BR-VPx) + GS + G(B + AB)u the equivalent control becomes u eq [G(B + AB)]~ 1 [G(AA + BR~ 1 B T P)x + GS]. (10) Substituting (10) into (1) and considering Assumption3, the ideal sliding mode dynamics becomes 144 Robust Control, Theory and Applications x = (A + AA)x - (B + AB)[G(B + AB)]~ 1 [G(AA + BR~ 1 B T P)x + G8] + 8 = (A + BAA)x - (B + BAB)[G(B + BAB)] _1 [GB(AAx + 8) + GBR _1 B T Px] + B£ (11) = (A-BR~ 1 B T P)x Comparing equation (11) with equation (7), we can see that they have the same form. So the sliding mode is asymptotically stable. Furthermore, it can be seen from (11) that the sliding mode is robust to uncertainties which satisfying matching conditions. So we call (8) a robust optimal sliding surface. 2.2.2 Design of sliding mode control law To ensure the reachability of sliding mode in finite time, we chose the sliding mode control law as follows: u(t) = u c (t) + u d (t), u c (t) = -R- 1 B T Px(t), (12) u d (t) = -(GBy 1 (rj + ro \\GB\\ + ri \\GB\\ \\x(t)\\) sgn(s). Where/7>0, u c (t) is the continuous part, used to stabilize and optimize the nominal system; u d (t) is the discontinuous part, which provides complete compensation for uncertainties of system (1). Let's select a quadratic performance index as follows: lit) = \\l[x T (t)Qx(t) + u T c (t)Ru c (t)\iL (13) where the meanings of Q and R are as the same as that in (4). Theorem 1. Consider uncertain linear system (1) with Assumptions 1-4. Let u and sliding surface be given by (12) and (8), respectively. The control law (12) can force the system trajectories with arbitrarily given initial conditions to reach the sliding surface in finite time and maintain on it thereafter. Proof. Choosing V = (l/2)s T s as a lyapunov function, and differentiating this function with respect to t and considering Assumptions 1-4, we have V = s T s = s T {G[(A + AA)x + (B + AB)u + 8]- G(A - BR~ 1 B T P)x} = s T {GBu + GB8 + GBR'VPx} = s T {-GBR~ 1 B T Px -(rj + ro \\GB\\ + y 1 ||GB||x|)sgn(s) + GB8 + GBR _1 B t Pjc} = s T {-(n + r \\GB\\ + n \\GB\\ \\x\\) sgn(s) + GB8} = ~#li "(^ol GB l + nl GB IIMI)INIi +sTgb $ * -#li - (ft \pB\\ + n |gs||MI)INIi + fro \pB\\ + n |gb||x|)|s| where |»| denotes 1-norm. Noting the fact that |s|L > |s|, we get V = s T s<-r]\\s\\ (14) This implies that the sliding mode control law we have chosen according to (12) could ensure the trajectories which start from arbitrarily given points be driven onto the sliding Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 145 surface (8) in finite time and would not leave it thereafter despite uncertainties. The proof is complete. Conclusion 1. The uncertain system (1) with the integral sliding surface (8) and the control law (12) achieves global sliding mode, and the performace index (13) is minimized. So the system designed is global robust and optimal. 2.3 Application to electrical servo drive The speed and position electrical servo drive systems are widely used in engineering systems, such as CNC machines, industrial robots, winding machines and etc. The main properties required for servo systems include high tracking behavior, no overshoot, no oscillation, quick response and good robustness. In general, with the implementation of field-oriented control, the mechanical equation of an induction motor drive or a permanent synchronous motor drive can be described as J m 0(t) + B m 0(t) + T d =T e (15) where 6 is the rotor position; J m is the moment of inertia; B m is the damping coefficient; T d denotes the external load disturbance, nonlinear friction and unpredicted uncertainties; T e represents the electric torque which defined as -K t i (16) where K t is the torque constant and i is the torque current command. Define the position tracking error e(t) = d (t) - 6{t) , where d (t) denotes the desired position, and let x a (f) = e(t) , x 2 (t) = x 1 (t) , u = i , then the error state equation of an electrical servo drive can be described as 1 J m U + (17) _*2_ = "0 1 -**- Jm _ X 2_ + "0 Jm _ U + " " 1 _Jm _ Supposing the desired position is a step signal, the error state equation can be simplified as (18) The parameters of the servo drive model in the nominal condition with T d = ONm are (Lin & Chou, 2003): / = 5.77 xlO" 2 Nms 2 , B = 8.8xlO" 3 Nms/rad, K t = 0.667 Nm/A. The initial condition is x(0) = [l 0] . To investigate the effectiveness of the proposed controller, two cases with parameter variations in the electrical servo drive and load torque disturbance are considered here. Case 1: /,„ = /,„ , B m = B m , T d = l(t - 10) Nm - l(f - 13)Nm . 146 Robust Control, Theory and Applications Case 2: J m =3J m , B n B n = 0. The optimal controller and the optimal robust SMC are designed, respectively, for both cases. The optimal controller is based on the nominal system with a quadratic performance index (4). Here Q- l o' 1 R = l In Case 1, the simulation results by different controllers are shown in Fig. 1. It is seen that when there is no disturbance ( t < 10s ), both systems have almost the same performance. 1 1 0.8 o I 0.6 04 /\ V - / "" / 1 T — Robust Optimal SMC -— Optimal Control 0.2 n ! 5 10 15 time(sec) (a) Position responses 0.6 _ 0.5 r C 0A 7 0.3 0.2 1 I Robust Optimal SMC Optimal Control n 5 10 15 20 time(sec) (b) Performance indexes Fig. 1. Simulation results in Case 1 But when a load torque disturbance occurs at f = (10 ~ 13)s , the position trajectory of optimal control system deviates from the desired value, nevertheless the position trajectory of the robust optimal SMC system is almost not affected. In Case 2, the simulation results by different controllers are given in Fig.2. It is seen that the robust optimal SMC system is insensitive to the parameter uncertainty, the position trajectory is almost as the same as that of the nominal system. However, the optimal control Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 147 system is affected by the parameter variation. Compared with the nominal system, the position trajectory is different, bigger overshoot and the relative stability degrades. In summery, the robust optimal SMC system owns the optimal performance and global robustness to uncertainties. 0.8 S 1 0.6 0.4 0.2 ^ , ( ji I i if r r Robust Optimal SMC Optimal Control Jr 1/ * 5 10 time(sec) (a) Position responses 15 20 Robust Optimal SMC Optimal Control 10 time(sec) 15 (b) Performance indexes Fig. 2. Simulation results in Case 2 2.4 Conclusion In this section, the integral sliding mode control strategy is applied to robustifying the optimal controller. An optimal robust sliding surface is designed so that the initial condition is on the surface and reaching phase is eliminated. The system is global robust to uncertainties which satisfy matching conditions and the sliding motion minimizes the given quadratic performance index. This method has been adopted to control the rotor position of an electrical servo drive. Simulation results show that the robust optimal SMCs are superior to optimal LQR controllers in the robustness to parameter variations and external disturbances. 148 Robust Control, Theory and Applications 3. Optimal sliding mode control for uncertain nonlinear system In the section above, the robust optimal SMC design problem for a class of uncertain linear systems is studied. However, nearly all practical systems contain nonlinearities, there would exist some difficulties if optimal control is applied to handling nonlinear problems (Chiou & Huang, 2005; Ho, 2007, Cimen & Banks, 2004; Tang et al, 2007). In this section, the global robust optimal sliding mode controller (GROSMC) is designed based on feedback linearization for a class of MIMO uncertain nonlinear system. 3.1 Problem formulation Consider an uncertain affine nonlinear system in the form of x = f(x) + g(x)u + d(t,x), y = H(x), [ ] where x e R n is the state, u e R m is the control input, and f(x) and g(x) are sufficiently smooth vector fields on a domain D czR n .Moreover, state vector x is assumed available, H(x) is a measured sufficiently smooth output function and H(x) = (h lf --- ,h m ) T • d(i,x) is an unknown function vector, which represents the system uncertainties, including system parameter variations, unmodeled dynamics and external disturbances. Assumption 5. There exists an unknown continuous function vector 8{t,x) such that d(t,x) can be written as d(t,x) = g(x)S(t,x). This is called matching condition. Assumption 6. There exist positive constants y and y x , such that \\S(t,x)\\<y + ri \\x\\ where the notation ||| denotes the usual Euclidean norm. By setting all the uncertainties to zero, the nominal system of the uncertain system (19) can be described as x = f(x) + g(x)u , y = H(x). { } The objective of this paper is to synthesize a robust sliding mode optimal controller so that the uncertain affine nonlinear system has not only the optimal performance of the nominal system but also robustness to the system uncertainties. However, the nominal system is nonlinear. To avoid the nonlinear TPBV problem and approximate linearization problem, we adopt the feedback linearization to transform the uncertain nonlinear system (19) into an equivalent linear one and an optimal controller is designed on it, then a GROSMC is proposed. 3.2 Feedback linearization Feedback linearization is an important approach to nonlinear control design. The central idea of this approach is to find a state transformation z = T(x) and an input transformation Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 149 u = u(x,v) so that the nonlinear system dynamics is transformed into an equivalent linear time-variant dynamics, in the familiar form z = Az + Bv , then linear control techniques can be applied. Assume that system (20) has the vector relative degree \j\,-'J m According to relative degree definition, we have and r-i Vi (*). ]} f h if 0<fc<r f -l and the decoupled matrix LAL}-\) - L.(Uy\) (21) E(x) = (e, h^f'K) L g„( L /"X) is nonsingular in some domain Vx e X . Choose state and input transformations as follows: zi=T i j (x) = L j f h i ,i = lr-,m;j = 0,lr-,r i -l (22) (23) u = E~ 1 (x)[v-K(x)], where K(x) = (Iljh^,--- ,L r ?/z m ) T , v is an equivalent input to be designed later. The uncertain nonlinear system (19) can be transformed into m subsystems; each one is in the form of (24) So system (19) can be transformed into the following equivalent model of a simple linear form: 10- • 1- • 0- • 1 0- • r o roi Z; 1 *i + v { + Si- 1 V z i 1 Vf\ z(t) = Az(t) + Bv(t) + co(t,z) (25) where zeR n , v g R m are new state vector and input, respectively. A e R nxn and B e R nxm are constant matrixes, and ( A, B) are controllable. &>(£,z)eR n is the uncertainties of the equivalent linear system. As we can see, co(t,z) also satisfies the matching condition, in other words there exists an unknown continuous vector function cb(t,z) such that co(t,z) = Bo>(t,z) . 1 50 Robust Control, Theory and Applications 3.3 Design of GROSMC 3.3.1 Optimal control for nominal system The nominal system of (25) is z(t) = Az(t) + Bv(t). (26) For (26), let v = v and v can minimize a quadratic performance index as follows: / = i{ o V(f)Qz« + v T (t)Rv (t)]dt (27) where Q e R nxn is a symmetric positive definite matrix, R e R mxm is a positive definite matrix. According to optimal control theory, the optimal feedback control law can be described as v (t) = -R- 1 B T Pz(t) (28) with P the solution of the matrix Riccati equation PA + A T P - PBR^B 7 ? + Q = 0. (29) So the closed-loop dynamics is z(t) = (A-BR- 1 B T P)z(t). (30) The closed-loop system is asymptotically stable. The solution to equation (30) is the optimal trajectory z*(t) of the nominal system with optimal control law (28). However, if the control law (28) is applied to uncertain system (25), the system state trajectory will deviate from the optimal trajectory and even the system becomes unstable. Next we will introduce integral sliding mode control technique to robustify the optimal control law, to achieve the goal that the state trajectory of uncertain system (25) is the same as that of the optimal trajectory of the nominal system (26). 3.3.2 The optimal sliding surface Considering the uncertain system (25) and the optimal control law (28), we define an integral sliding surface in the form of s(t) = G[z(t) - z(0)] - Gj*(A - BR- 1 B T P)z(r)dr (31) where G e R mxn f which satisfies that GB is nonsingular, z(0) is the initial state vector. Differentiating (31) with respect to t and considering (25), we obtain s(t) = Gz(t) - G(A - BR~ 1 B T P)z(t) = G[Az(t) + Bv(t) + (o(i, z)\ - G(A - BR~ 1 B T P)z(t) (32) = GBv(t) + GBR-^Pzit) + Gco(i,z) Let s(t ) = , the equivalent control becomes Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 151 (33) (34) v eq (t) = -(GB)" 1 [GBR-^Pz^t) + GcD{t,z)\ Substituting (33) into (25), the sliding mode dynamics becomes z = Az- B{GB)~ 1 (GBR-^Pz + Gco) + co = Az-BR- 1 B T Pz-B(GBy 1 Gco + co = Az- BR~ 1 B T Pz - B(GB)" 1 GBcb + Bcb = (A-BR~ 1 B T P)z Comparing (34) with (30), we can see that the sliding mode of uncertain linear system (25) is the same as optimal dynamics of (26), thus the sliding mode is also asymptotically stable, and the sliding motion guarantees the controlled system global robustness to the uncertainties which satisfy the matching condition. We call (31) a global robust optimal sliding surface. Substituting state transformation z = T(x) into (31), we can get the optimal switching function s(x,t) in the x -coordinates. 3.3.3 The control law After designing the optimal sliding surface, the next step is to select a control law to ensure the reachability of sliding mode in finite time. Differentiating s(x,t) with respect to t and considering system (20), we have ds . ds ds , r/ . / v v ds s = ^ x + ^ = ^fW + ^ x » + T7- dx dt dx dt Let s = , the equivalent control of nonlinear nominal system (20) is obtained -i-l r "«,(') = g(*) ds r . x ds — f(x) + — dx dt (35) (36) Considering equation (23), we have u = E 1 (x)[v -K(x)] . Now, we select the control law in the form of M disW : U(t] = M con(0 + "dis(0/ M can(*) = - ds T-8( x ) _dx -1 ds r/ . ds — f(x) + — , _dx JK } dt] --- ds -z-g(x) _dx -l (7 + (/o + rilMI) ds T-g(*) OX (37) )sg n ( s )> where sgn(s) = [sgn(s a ) sgn(s 2 ) ••• sgn(s m )] and //>0. u con (t) and u dis (t) denote continuous part and discontinuous part of u(t) , respectively. The continuous part u con (t) , which is equal to the equivalent control of nominal system (20), is used to stabilize and optimize the nominal system. The discontinuous part u dis (t) provides the complete compensation of uncertainties for the uncertain system (19). Theorem 2. Consider uncertain affine nonlinear system (19) with Assumputions 5-6. Let u and sliding surface be given by (37) and (31), respectively. The control law can force the system trajectories to reach the sliding surface in finite time and maintain on it thereafter. 152 Robust Control, Theory and Applications Proof. Utilizing V = (1 / 2)s s as a Lyapunov function candidate, and taking the Assumption 5 and Assumption 6, we have tV T • T / dS . r j. OS. ox ot -AtJ- ds . ds — / + — dx di r + (ro+ri\\x 1) as ax sgn(s) as t 5s 1 dx dt\ -4 7 + (ro+riH)-^ || 5x } ds ' sgn(s)Us T — ^ = -/7|H| 1 -(^o+riH) ds 1 dx || |s| 1+ s T !V (38) 1 dx ^-^INIi-^o + riWI) as ax II II & 1 fax INI HI * < — 77||s|| !-(r o + riH) as dx ll s li + (^o + ri WD as dx IN where |»| denotes the 1-norm. Noting the fact that |s| > |s|, we get V = s s<-r/\\s\\<0 ,for|s|U0. (39) This implies that the trajectories of the uncertain nonlinear system (19) will be globally driven onto the specified sliding surface s = despite the uncertainties in finite time. The proof is complete. From (31), we have s(0) = , that is the initial condition is on the sliding surface. According to Theorem2, we know that the uncertain system (19) with the integral sliding surface (31) and the control law (37) can achieve global sliding mode. So the system designed is global robust and optimal. 3.4 A simulation example Inverted pendulum is widely used for testing control algorithms. In many existing literatures, the inverted pendulum is customarily modeled by nonlinear system, and the approximate linearization is adopted to transform the nonlinear systems into a linear one, then a LQR is designed for the linear system. To verify the effectiveness and superiority of the proposed GROSMC, we apply it to a single inverted pendulum in comparison with conventional LQR. The nonlinear differential equation of the single inverted pendulum is . _ g sin x 1 - amhx\ sin x 1 cos x 1 + au cos x 1 X2 — 2 L(4 / 3-amcos x x ) ■d(t), (40) where x 1 is the angular position of the pendulum (rad) , x 2 is the angular speed (rad/s) , M is the mass of the cart, m and L are the mass and half length of the pendulum, respectively, u denotes the control input, g is the gravity acceleration, d(t) represents the external disturbances, and the coefficient a = m / (M + m) . The simulation parameters are as follows: M = 1 kg , m = 0.2 kg , L = 0.5 m , g = 9.8 m/s 2 , and the initial state vector is x(0) = [-tt /IS 0] T . Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 153 Two cases with parameter variations in the inverted pendulum and external disturbance are considered here. Case 1: The m and L are 4 times the parameters given above, respectively. Fig. 3 shows the robustness to parameter variations by the suggested GROSMC and conventional LQR. Case 2: Apply an external disturbance d(t) = 0.01 sin 2t to the inverted pendulum system at t = 9s. Fig. 4 depicts the different responses of these two controllers to external disturbance. 0.02 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 ^ -0.18 1 /-' III m=0.2 kg,L=0.5 m m=0.8 kg,L=2 m / 1 i ; ; ; ; ii ii 0.02 -0.02 ® -0.04 § -0.06 o -0.08 | -0.1 en < -0.12 -0.14 -0.16 -0.18 / A" — m=0.2 kg,L=0.5 m m=0.8 kg,L=2 m l_ _L I I t(s) t(s) (a) By GROSMC (b) By Conventional LQR. Fig. 3. Angular position responses of the inverted pendulum with parameter variation 1 ^T^ -z--'^^ -0.02 L _ — Optimal Control -0 04 \< -X GROSMC -0.06 ;'l -0.08 <> V -0.1 I 1 -0.12 11 -0.14 -0.16 10 t(s) Fig. 4. Angular position responses of the inverted pendulum with external disturbance. From Fig. 3 we can see that the angular position responses of inverted pendulum with and without parameter variations are exactly same by the proposed GROSMC, but the responses are obviously sensitive to parameter variations by the conventional LQR. It shows that the proposed GROSMC guarantees the controlled system complete robustness to parameter variation. As depicted in Fig. 4, without external disturbance, the controlled system could be driven to the equilibrium point by both of the controllers at about t = 2s. However, when the external disturbance is applied to the controlled system at t = 9s, the inverted pendulum system could still maintain the equilibrium state by GROSMC while the LQR not. 154 Robust Control, Theory and Applications The switching function curve is shown in Fig. 5. The sliding motion occurs from the beginning without any reaching phase as can be seen. Thus, the GROSMC provides better features than conventional LQR in terms of robustness to system uncertainties. 0.5 0.4 0.2 0.1 e V) -0.1 -0.2 -0.3 -0.4 -0.5 — Sliding Surface t(s) Fig. 5. The switching function s(t) 3.5 Conclusion In this section, the exact linearization technique is firstly adopted to transform an uncertain affine nonlinear system into a linear one. An optimal controller is designed to the linear nominal system, which not only simplifies the optimal controller design, but also makes the optimal control applicable to the entire transformation region. The sliding mode control is employed to robustfy the optimal regulator. The uncertain system with the proposed integral sliding surface and the control law achieves global sliding mode, and the ideal sliding dynamics can minimized the given quadratic performance index. In summary, the system designed is global robust and optimal. 4. Optimal sliding mode tracking control for uncertain nonlinear system With the industrial development, there are more and more control objectives about the system tracking problem (Ouyang et al., 2006; Mauder, 2008; Smolders et al., 2008), which is very important in control theory synthesis. Taking the robot as an example, it is often required to follow some special trajectories quickly as well as provide robustness to system uncertainties, including unmodeled dynamics, internal parameter variations and external disturbances. So the main tracking control problem becomes how to design the controller, which can not only get good tracking performance but also reject the uncertainties effectively to ensure the system better dynamic performance. In this section, a robust LQR tracking control based on intergral sliding mode is proposed for a class of nonlinear uncertain systems. 4.1 Problem formulation and assumption Consider a class of uncertain affine nonlinear systems as follows: x = f(x) + Af(x) + g(x)[u + S(x,t,u)] y = h(x) (41) Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 155 where x e R n is the state vector, u e R m is the control input with m = 1 , and y e jR is the system output. f(x) , g(x) , A/(x) and h(x) are sufficiently smooth in domain D czR n . 8{x,t,u) is continuous with respect to t and smooth in (x,u) . Af(x) and 5{x,t,u) represent the system uncertainties, including unmodelled dynamics, parameter variations and external disturbances. Our goal is to design an optimal LQR such that the output y can track a reference trajectory y r (t) asymptotically, some given performance criterion can be minimized, and the system can exhibit robustness to uncertainties. Assumption 7. The nominal system of uncertain affine nonlinear system (41), that is x = f(x) + g(x)u ^ [y = H x ) has the relative degree p in domain D and p = n . Assumption 8. The reference trajectory y T (t) and its derivations y[ l \t) (i = l,---,n)can be obtained online, and they are limited to all t > . While as we know, if the optimal LQR is applied to nonlinear systems, it often leads to nonlinear TPBV problem and an analytical solution generally does not exist. In order to simplify the design of this tracking problem, the input-output linearization technique is adopted firstly. Considering system (41) and differentiating y , we have y {k) =L k f h(x), 0<k<n-l y (n) = L n f h(x) + LyLy^ix) + h^\(x)\u + 8(x,t,u)\. According to the input-out linearization, choose the following nonlinear state transformation z = T(x) = [h(x) .- L}- a /z(x)] T . (43) So the uncertain affine nonlinear system (40) can be written as Zi=z i+1 , i = !,-•• f n-l z n = L n f h(x) + L^L^hix) + h^\(x)\u + 8(x f t f u)\. Define an error state vector in the form of z -v (n_1) = z-K, where ^R = [y • • • y^ n ^ 1 -By this variable substitution e = z - $R , the error state equation can be described as follows: e i ~ e i+i' i = l,-',n-l e n = L n f h(x) + L^ x h(x) + h g ^\{x)u(i) + L g L n f 1 h(x)S(x / 1, u) - y[ n) (t). 156 Robust Control, Theory and Applications Let the feedback control law be selected as u(t) = -L n f h(x) + v(t) + y^(t) Lff^x) (44) The error equation of system (40) can be given in the following forms: m- 1 ... 1 ... ... 1 it) + "0" + v(t) + Af L n f 1 h(x)_ 1 LL n f 1 h(x)S(x / t / u) (45) Therefore, equation (45) can be rewritten as e(t) = Ae(t) + AA + Bv(t) + AS. where (46) "0 1 o ... 0" "0" 1 ••• A = o o o ... 1 , B = 1 ' , AS = v* y 1 h(x)_ I g iy\{: t)S (x,t,u) AA-- As can be seen, e g R n is the system error vector, v e R is a new control input of the transformed system. A e R nxn and B e R nxm are corresponding constant matrixes. AA and A£ represent uncertainties of the transformed system. Assumption 9. There exist unknown continuous function vectors of appropriate dimensions AA and A^ , such that AA = BAA , A£ = BAS Assumption 10. There exist known constants a m , b m such that H<a m ,|A^|<^ Now, the tracking problem becomes to design a state feedback control law v such that e -^ asymptotically. If there is no uncertainty, i.e. S(t,e) = , we can select the new input as v = -Ke to achieve the control objective and obtain the closed loop dynamics e = (A- BK)e . Good tracking performance can be achieved by choosing K using optimal Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 157 control theory so that the closed loop dynamics is asymptotically stable. However, in presence of the uncertainties, the closed loop performance may be deteriorated. In the next section, the integral sliding mode control is adopted to robustify the optimal control law. 4.2 Design of optimal sliding mode tracking controller 4.2.1 Optimal tracking control of nominal system. Ignoring the uncertainties of system (46), the corresponding nominal system is e(t) = Ae(t) + Bv(t). (47) For the nominal system (47), let v = v and v can minimize the quadratic performance index as follows: I = l^[e\t)Qe(t) + v T (t)Rv (t)]dt (48) where Q e R nxn is a symmetric positive definite matrix, R e R mxm (here m = 1 ) is a positive definite matrix. According to optimal control theory, an optimal feedback control law can be obtained as: v (t) = -R- 1 B T Pe(t) (49) with P the solution of matrix Riccati equation PA + A T P - PBRT^P + Q = 0. So the closed-loop system dynamics is e(t) = (A-BR- 1 B T P)e(t). (50) The designed optimal controller for system (47) is sensitive to system uncertainties including parameter variations and external disturbances. The performance index (48) may deviate from the optimal value. In the next part, we will use integral sliding mode control technique to robustify the optimal control law so that the uncertain system trajectory could be same as nominal system. 4.2.2 The robust optimal sliding surface. To get better tracking performance, an integral sliding surface is defined as s(e, t) = Ge(t) -Gf(A- BR^B 1 P)e{r)dr - Ge(Q), (51) where G e R mxn i s a constant matrix which is designed so that GB is nonsingular. And e(0) is the initial error state vector. Differentiating (51) with respect to t and considering system (46), we obtain s(e, t) = Ge(t) - G(A - BR~ 1 B T P)e(t) = G[Ae(t) + AA + Bv(t) + AS] - G(A - BR~ 1 B T P)e(t) (52) -i d t GBv(t) + GBR-'B L Pe(t) + G(AA + AS). 1 58 Robust Control, Theory and Applications Let s(e,t) = , the equivalent control can be obtained by v eq (t) = -(GB)- 1 [GBR- 1 B T Pe(0 + G(AA + AS)]. (53) Substituting (53) into (46), and considering Assumption 10, the ideal sliding mode dynamics becomes e(t) = Ae(t) + AA + Bv (t) + AS = Ae(t) + AA- B(GB)~ 1 [GBR~ 1 B Y Pe(t) + G(AA + AS)] + AS = (A - BR~ 1 B T P)e(t) - B(GB)~ 1 G[ AA + AS] + AA + AS (54) = (A - BR~ 1 B T P)e(t) - B(GB)~ 1 GB(AA + AS) + B(AA + AS) = (A-BR~ 1 B T P)e(t). It can be seen from equation (50) and (54) that the ideal sliding motion of uncertain system and the optimal dynamics of the nominal system are uniform, thus the sliding mode is also asymptotically stable, and the sliding mode guarantees system (46) complete robustness to uncertainties. Therefore, (51) is called a robust optimal sliding surface. 4.2.3 The control law. For uncertain system (46), we propose a control law in the form of v(t) = v c (t) + v d (t), v c (t) = -R- 1 B T Pe(t), (55) v d (t) = -(GB)- 1 [ks + e s S n(s)]. where v c is the continuous part, which is used to stabilize and optimize the nominal system. And v d is the discontinuous part, which provides complete compensation for system uncertainties, sgn(s) = [sgn(s a ) ••• sgn(s m )] . k and s are appropriate positive constants, respectively. Theorem 3. Consider uncertain system (46) with Assumption9-10. Let the input v and the sliding surface be given as (55) and (51), respectively. The control law can force system trajectories to reach the sliding surface in finite time and maintain on it thereafter if s>(a m +d m )\\GB\\. Proof: Utilizing V = (1 / 2)s T s as a Lyapunov function candidate, and considering Assumption 9-10, we obtain V = s T s = s T [Ge(t) - G(A - BR~ 1 B T P)e(t)] = s T {G[Ae(t) + AA + Bv(t) + AS] - G(A - BR- 1 B T P)e(t)] = s T [GAA - GBR^Peit) -(ks + s sgn(s)) + GAS + GBR^Pe^ = s T {- [ks + s sgn(s)] + GAA + GAS) = -k ||s|| a - e \\s\\ + s T (GAA + GAS) ^ -^ll s lli - ^ll s ll + (^ m + ^ m )||GS||||^|| ^ -^ll^ll - [^ - (^ m + ^ m )|| GB ll]ll s ll where |»| denotes the 1-norm. Note the fact that for any |s| * , we have |s| > \\s\\ . If ^K+^)|GB|,then Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 159 V --s T s<- -^ IpIj +||G|| |4|s||<-(f-|G|W)|s|<0. (56) This implies that the trajectories of uncertain system (46) will be globally driven onto the specified sliding surface s(e,t) = in finite time and maintain on it thereafter. The proof is completed. From (51), we have s(0) = , that is to say, the initial condition is on the sliding surface. According to Theorem3, uncertain system (46) achieves global sliding mode with the integral sliding surface (51) and the control law (55). So the system designed is global robust and optimal, good tracking performance can be obtained with this proposed algorithm. 4.3 Application to robots. In the recent decades, the tracking control of robot manipulators has received a great of attention. To obtain high-precision control performance, the controller is designed which can make each joint track a desired trajectory as close as possible. It is rather difficult to control robots due to their highly nonlinear, time-varying dynamic behavior and uncertainties such as parameter variations, external disturbances and unmodeled dynamics. In this section, the robot model is investigated to verify the effectiveness of the proposed method. A 1-DOF robot mathematical model is described by the following nonlinear dynamics: 1 r- ~ C(q,q) q - G(q) + 1 T - 1 M(q)\ w_ [M(q)_ [M(q)\ [M(q)\ d(t), (57) where q, q denote the robot joint position and velocity, respectively, r is the control vector of torque by the joint actuators, m and / are the mass and length of the manipulator arm, respectively. d(t) is the system uncertainties. C(q,q) = 0. 03 cos(q), G(q) = mglcos(q), M(q) = 0.1 + 0.06sin(^). The reference trajectory is y T (t) = sin^-f . According to input-output linearization technique, choose a state vector as follows: V .4. Define an error state vector of system (57) as e = \e x e 2 ] = [q-y T 4 — 3/r ] / anc ^ ^ the control law r = (v + y r )M(q) + C(q,q)q + G(q) . So the error state dynamic of the robot can be written as: V "0 1* V + "0" v- L^2_ L° °J L^2_ 1 [l/M(q)\ d(t) (58) Choose the sliding mode surface and the control law in the form of (51) and (55), respectively, and the quadratic performance index in the form of (48). The simulation parameters are as follows: ra = 0.02, g = 9.8, 1 = 0.5, d(t) = 0.5 sin27rt, A: = 18, s = 6, l~10 2l t G = [0 l], Q= t R = 1 . The initial error state vector is e = [0.5 0] . The tracking responses of the joint position ^and its velocity are shown in Fig. 6 and Fig. 7, respectively. The control input x is displayed in Fig. 8. From Fig. 6 and Fig. 7 it can be seen that the position error can reach the equilibrium point quickly and the position track the 160 Robust Control, Theory and Applications reference sine signal y r well. Simulation results show that the proposed scheme manifest good tracking performance and the robustness to parameter variations and the load disturbance. 4.4 Conclusions In order to achieve good tracking performance for a class of nonlinear uncertain systems, a sliding mode LQR tracking control is developed. The input-output linearization is used to transform the nonlinear system into an equivalent linear one so that the system can be handled easily. With the proposed control law and the robust optimal sliding surface, the system output is forced to follow the given trajectory and the tracking error can minimize the given performance index even if there are uncertainties. The proposed algorithm is applied to a robot described by a nonlinear model with uncertainties. Simulation results illustrate the feasibility of the proposed controller for trajectory tracking and its capability of rejecting system uncertainties. Fig. 6. The tracking response of q - reference speed — speed response Fig. 7. The tracking response of q Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 161 1.5 1 I n A, A, , A An a. ^ 0.5 I ill | ; 1 1 1 J 'l lj\ \ i iljl 1 i'l 1 ■Vi -0.5 1 1 lit ,i 'ill il' I'l 1 ' \ n 1 ' kl'l ' 1 1 -1 f" ||| 1 1 -1.5 | | 1 1 1 II I -2 i f f f I 1 I -2.5 Fig. 8. The control input r 5. Acknowledgements This work is supported by National Nature Science Foundation under Grant No. 60940018. 6. References Basin, M.; Rodriguez-Gonzaleza, J.; Fridman, L. (2007). Optimal and robust control for linear state-delay systems. Journal of the Franklin Institute. Vol.344, pp.830-845. Chen, W. D.; Tang, D. Z.; Wang, H. T. (2004). Robust Tracking Control Of Robot M anipulators Using Backstepping. 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Introduction It is well known that many engineering control systems such as conventional oil-chemical industrial processes, nuclear reactors, long transmission lines in pneumatic, hydraulic and rolling mill systems, flexible joint robotic manipulators and machine-tool systems, jet engine and automobile control, human-autopilot systems, ground controlled satellite and networked control and communication systems, space autopilot and missile-guidance systems, etc. contain some time-delay effects, model uncertainties and external disturbances. These processes and plants can be modeled by some uncertain dynamical systems with state and input delays. The existence of time-delay effects is frequently a source of instability and it degrades the control performances. The stabilization of systems with time-delay is not easier than that of systems without time-delay. Therefore, the stability analysis and controller design for uncertain systems with delay are important both in theory and in practice. The problem of robust stabilization of uncertain time-delay systems by various types of controllers such as PID controller, Smith predictor, and time-delay controller, recently, sliding mode controllers have received considerable attention of researchers. However, in contrast to variable structure systems without time-delay, there is relatively no large number of papers concerning the sliding mode control of time-delay systems. Generally, stability analysis can be divided into two categories: delay-independent and delay-dependent. It is worth to mention that delay-dependent conditions are less conservative than delay-independent ones because of using the information on the size of delays, especially when time-delays are small. As known from (Utkin, 1977)-(Jafarov, 2009) etc. sliding mode control has several useful advantages, e.g. fast response, good transient performance, and robustness to the plant parameter variations and external disturbances. For this reason, now, sliding mode control is considered as an efficient tool to design of robust controllers for stabilization of complex systems with parameter perturbations and external disturbances. Some new problems of the sliding mode control of time-delay systems have been addressed in papers (Shyu & Yan, 1993)-(Jafarov, 2005). Shyu and Yan (Shyu & Yan, 1993) have established a new sufficient condition to guarantee the robust stability and ^-stability for uncertain systems with single time-delay. By these conditions a variable structure controller is designed to stabilize the time-delay systems with uncertainties. Koshkoei and Zinober (Koshkouei & Zinober, 1996) have designed a new 1 64 Robust Control, Theory and Applications sliding mode controller for MIMO canonical controllable time-delay systems with matched external disturbances by using Lyapunov-Krasovskii functional. Robust stabilization of time-delay systems with uncertainties by using sliding mode control has been considered by Luo, De La Sen and Rodellar (Luo et al., 1997). However, disadvantage of this design approach is that, a variable structure controller is not simple. Moreover, equivalent control term depends on unavailable external disturbances. Li and DeCarlo (Li & De Carlo, 2003) have proposed a new robust four terms sliding mode controller design method for a class of multivariable time-delay systems with unmatched parameter uncertainties and matched external disturbances by using the Lyapunov-Krasovskii functional combined by LMFs techniques. The behavior and design of sliding mode control systems with state and input delays are considered by Perruquetti and Barbot (Perruquetti & Barbot, 2002) by using Lyapunov-Krasovskii functional. Four-term robust sliding mode controllers for matched uncertain systems with single or multiple, constant or time varying state delays are designed by Gouaisbaut, Dambrine and Richard (Gouisbaut et al., 2002) by using Lyapunov-Krasovskii functionals and Lyapunov- Razumikhin function combined with LMFs techniques. The five terms sliding mode controllers for time- varying delay systems with structured parameter uncertainties have been designed by Fridman, Gouisbaut, Dambrine and Richard (Fridman et al., 2003) via descriptor approach combined by Lyapunov-Krasovskii functional method. In (Cao et al., 2007) some new delay-dependent stability criteria for multivariable uncertain networked control systems with several constant delays based on Lyapunov-Krasovskii functional combined with descriptor approach and LMI techniques are developed by Cao, Zhong and Hu. A robust sliding mode control of single state delayed uncertain systems with parameter perturbations and external disturbances is designed by Jafarov (Jafarov, 2005). In survey paper (Hung et al., 1993) the various type of reaching conditions, variable structure control laws, switching schemes and its application in industrial systems is reported by J. Y.Hung, Gao and J.C.Hung. The implementation of a tracking variable structure controller with boundary layer and feed-forward term for robotic arms is developed by Xu, Hashimoto, Slotine, Arai and Harashima(Xu et al., 1989). A new fast-response sliding mode current controller for boost-type converters is designed by Tan, Lai, Tse, Martinez-Salamero and Wu (Tan et al., 2007). By constructing new types of Lyapunov functionals and additional free- weighting matrices, some new less conservative delay-dependent stability conditions for uncertain systems with constant but unknown time-delay have been presented in (Li et al., 2010) and its references. Motivated by these investigations, the problem of sliding mode controller design for uncertain multi-input systems with several fixed state delays for delay-independent and delay-dependent cases is addressed in this chapter. A new combined sliding mode controller is considered and it is designed for the stabilization of perturbed multi-input time-delay systems with matched parameter uncertainties and external disturbances. Delay- independent/ dependent stability and sliding mode existence conditions are derived by using Lyapunov-Krasovskii functional and Lyapunov function method and formulated in terms of LMI. Delay bounds are determined from the improved stability conditions. In practical implementation chattering problem can be avoided by using saturation function (Hung et al, 1993), (Xu et al, 1989). Five numerical examples with simulation results are given to illustrate the usefulness of the proposed design method. Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 65 2. System description and assumptions Let us consider a multi-input state time-delay systems with matched parameter uncertainties and external disturbances described by the following state-space equation: x(t) = (A + AA )x(t) + (A 1 + AA x )x{t -h 1 ) + ... + (A N + AA N )x(t -h N ) + Bu(t) + Df(t), t > x(t) = 0(t), -h<t<0 (1) where x(t) e R n is the measurable state vector, u(t) eR m is the control input, A ,A 1 ,..,A N and B are known constant matrices of appropriate dimensions, with B of full rank, h = mdix[h 1 ,h 1 ,...,h N ],h l >0 , h lf h 2f ... f h N are known constant time-delays, (/){t) is a continuous vector-valued initial function in -h<t<0; AA ,AA 1/ ...,AA N and D are the parameter uncertainties, (/){t) is unknown but norm-bounded external disturbances. Taking known advantages of sliding mode, we want to design a simple suitable sliding mode controller for stabilization of uncertain time-delay system (1). We need to make the following conventional assumptions for our design problem. Assumption 1: a. (A ,B) is stabilizable; b. The parameter uncertainties and external disturbances are matched with the control input, i.e. there exist matrices E (t), E(t), E 1 (t), . . . , E N (t ) , such that: AA (t) = BE (t) ; A 1 (t) = BE 1 (t) ; ...,AA N (t) = BE N (t) ; D(t) = BE(t) (2) with norm-bounded matrices: max|zlE (f)| < a ; max|zlE 1 (f)| < a x ; ...,max|zlE N (f)| < a N \\E(t)\\ = a l/Wl^/o (3) where a^,a x ,a x ,...a n ,g and f are known positive scalars. The control goal is to design a combined variable structure controller for robust stabilization of time-delay system (1) with matched parameter uncertainties and external disturbances. 3. Control law and sliding surface To achieve this goal, we form the following type of combined variable structure controller: ^) = u Un (t) + u eq (t) + u vs (t) + u r (t) (4) where Ui m (t) = -Gx(t) (5) u eq (t) = -(CB)" 1 [CA x(t) + CA lX (t - V + ■ • ■ + CA N x(t -h N )] (6) 1 66 Robust Control, Theory and Applications u vs (t) = -[k \\x(t)\\ + k4x(t-h,)\\ + ...,+k N \\x(t-h N )\\]^ (7) "■- s m where k§,k\,--->k^ and 5 are the scalar gain parameters to be selected; Gis a design matrix; (CBy 1 is a non-singular m x m matrix. The sliding surface on which the perturbed time-delay system states must be stable is defined as a linear function of the undelayed system states as follows: s(t) = rCx(t) (9) where C is a m x n gain matrix of full rank to be selected; r is chosen as identity mxm matrix that is used to diagonalize the control. Equivalent control term (6) for non-perturbed time-delay system is determined from the following equations: s(t) = Cx(t) = CA x(t) + CA x x{t -h 1 ) + ... + CA N x(t -h N ) + CBu(t) = (10) Substituting (6) into (1) we have a non-perturbed or ideal sliding time-delay motion of the nominal system as follows: x(t) = Aox(t) + Aix(t-h 1 ) + ... + A N x(t-h N ) (H) where (CBy'C = G eq , A - BG eq A = Ac , A 1 - BG eq A x = M, ..., A N - BG eq A N = A N (12) Note that, constructed sliding mode controller consists of four terms: 1. The linear control term is needed to guarantee that the system states can be stabilized on the sliding surface; 2. The equivalent control term for the compensation of the nominal part of the perturbed time-delay system; 3. The variable structure control term for the compensation of parameter uncertainties of the system matrices; 4. The min-max or relay term for the rejection of the external disturbances. Structure of these control terms is typical and very simple in their practical implementation. The design parameters G,C,k Q ,k 1 ,...,k- N 8 of the combined controller (4) for delay- independent case can be selected from the sliding conditions and stability analysis of the perturbed sliding time-delay system. However, in order to make the delay-dependent stability analysis and choosing an appropriate Lyapunov-Krasovskii functional first let us transform the nominal sliding time- delay system (11) by using the Leibniz-Newton formula. Since x(t) is continuously differentiable for t > 0, using the Leibniz-Newton formula, the time-delay terms can be presented as: t t x(t-h 1 ) = x(t)- j* x(0)d0,...,x(t-h N ) = x(t)- j* x{6)d6 (13) t-\ t-h N Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 67 Then, the system (11) can be rewritten as _ _ _ l _ l x(t) = (Ao+Ai+... + A N )x(t)-Ai J x(0)d0-...-A N J x(0)d0 (14) t-h t t-h N Substituting again (11) into (14) yields: t x(t) = (A +A 1 +... + A N )x(t)-A 1 J [A o x(0) + A 1 x(0-h 1 ) + ... + A N x(0-h N )]l0 t-\ t -...-A N J [A o x(0) + A 1 x(0-h 1 ) + ... + A N x(0-h N )]l0 t-h N (15) t t t = (A +A 1 +... + A N )x(t)-A 1 A J x(0)d0-A\ J x(0-h 1 )d0-...-A 1 A N J x(0-h N )d0 t-\ t-hi t-\ t t t -...-A n Aq J x(0)d0-A N A 1 J x(0-h 1 )d0-...-A% } J x(0-h N )d0 t-h N t-h N t-h N Then in adding to (15) the perturbed sliding time-delay system with control action (4) or overall closed loop system can be formulated as: t t x(t) = (A + A l +... + A N )x(t)-A l A J x{0)d0-Al J x(0-h l )d0 t-h x t-\ t t t -...-A X A N J x(0-h N )d0-...-A N A o J x(0)d0-A N A l J x(0-h l )d0 t-\ t-h N t-h N t -...-A 2 N J x(0-h N )d0 + AA o x(t) (16) t-h N +AA { x(t -h l ) + ... + AA N x(t - h N ) -B[k \\x(t)\\ + ^||x^ - ^)|| + .., + k N \\x(t - h N )\\\^-BS^- + Df(t) where A =A -BG, the gain matrix G can be selected such that A has the desirable eigenvalues. The design parameters G,C,kQ,k 1 ,...,k N S of the combined controller (4) for delay- dependent case can be selected from the sliding conditions and stability analysis of the perturbed sliding time-delay system (16). 4. Robust delay-independent stabilization In this section, the existence condition of the sliding manifold and delay-independent stability analysis of perturbed sliding time-delay systems are presented. 4.1 Robust delay-independent stabilization on the sliding surface In this section, the sliding manifold is designed so that on it or in its neighborhood in different from existing methods the perturbed sliding time-delay system (1),(4) is globally 168 Robust Control, Theory and Applications asymptotically stable with respect to state coordinates. The perturbed stability results are formulated in the following theorem. Theorem 1: Suppose that Assumption 1 holds. Then the multivariable time-delay system (1) with matched parameter perturbations and external disturbances driven by combined controller (4) and restricted to the sliding surface s(t)=0 is robustly globally asymptotically delay-independent stable with respect to the state variables, if the following LMI conditions and parameter requirements are satisfied: H= ^^ iVL - u <0 (17) (18) (19) (20) where P,R 1 ,...R N are some symmetric positive definite matrices which are a feasible solution of LMI (17) with (18); A = A - BG in which a gain matrix G can be assigned by pole placement such that A has some desirable eigenvalues. Proof: Choose a Lyapunov-Krasovskii functional candidate as follows: A T P + PA +R^+... + R N PAi .. PA N {PM) T -R a .. (PAn) t .. -R» CB = B T PB > ^0 = ^O'^l = ^l'f-'f^N = a N s*h N t V = x T (t)Px(t) + ^ J x 1 \6)R i x{6)de i=lt-h: (21) The time-derivative of (21) along the state trajectories of time-delay system (1), (4) can be calculated as follows: V = 2x T (t)P[A x(t) + A x x(i -h 1 ) + ... + A N x(t -h N ) + AA x(t) + AA x x(i - \) + ... + AA N x(t -h N ) + Bu(t) + Df(t)] + x t (0^ 1 x(0-x t (^-/z 1 )^ 1 x(^-/z 1 ) + ... + x t (0^ n x(0-^ T (^-^ n )^n x (^-^n) - 2x T (t)PAox(t) + 2x T (t)PAix(t -h 1 ) + ... + 2x T (t)PA N x(t -h N ) + 2x T (t)PBE x(t) + 2x T (t)PBE 1 x(t-h 1 ) + ... + 2x T (t)PBE N x(t-h N ) - 2x T (t)PB[k \\x(t)\\ + k x \\x(t - \ )|| + .., + k N \\x(t -h N )^ S{t) s(t)\\ - 2x T (t)PBGx(t) - 2Sx T (t)PB /^ + 2x T (t)PBEf(t) s (0| + x T (t)(R 1 +... + R N )x(t)-x T (t-h 1 )R 1 x(t-h 1 )-...-x T (t-h N )R N x(t-h N ) Since x (t)PB = s (t) , then we obtain: Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 169 V<x T (t) A P + PA +R 1 . + R N ]x(t) -2x T (t)PA N x(t-h N ) + 2x T (t)PAix(t-h 1 ) -x T (t-h 1 )R 1 x(t-h 1 )-...-x T (t-h 1 )R N x(t-h N ) + 2s T (t)E x(t) + 2s T (f )Ei*(* -h 1 ) + ... + 2s T (t)E N x(t - h N ) + 2s T (t)Ef(t)-2s T (t)[k \\x(t)\\ + k4x(t HOI HOI x(t) x(i-\) _x(t-h N ] - [(7c - « )||^:(0|||k(0|| + (^o. - «a)||^(^ - /z x )|| ||s(0|| + --- + (^^ - ^^)||^(^ - ^^)|||k(0||] -(*-/o)H')l Since (17)-(20) hold, then (22) reduces to: T A T P + PA +R 1 +. . + R N PAi . .. PA N r X (t) (PMf -Ri ■ x{t-\) (PA N ) T o . .. -R N _ At- h N) (22) V<z T (t)Hz(t)<0 (23) where z T (t) = [x(t)x(t-h 1 )...x(t-h^)]. Therefore, we can conclude that the perturbed time-delay system (1), (4) is robustly globally asymptotically delay-independent stable with respect to the state coordinates. Theorem 1 is proved. 4.2 Existence conditions The final step of the control design is the derivation of the sliding mode existence conditions or the reaching conditions for the perturbed time-delay system (1),(4) states to the sliding manifold in finite time. These results are summarized in the following theorem. Theorem 2: Suppose that Assumption 1 holds. Then the perturbed multivariable time- delay system (1) states with matched parameter uncertainties and external disturbances driven by controller (4) converge to the siding surface s(t)=0 in finite time, if the following conditions are satisfied: k =a +g;k 1 =a 1 ;...;k N =a N Proof: Let us choose a modified Lyapunov function candidate as: V=h T (t)(CB)-'s(t) (24) (25) (26) The time-derivative of (26) along the state trajectories of time-delay system (1), (4) can be calculated as follows: 170 Robust Control, Theory and Applications V = s T (f)(CB)" 1 s(f) = s r (f)(CB)" 1 Ci:(f) = s T (t )(CB)~ 1 C [A x(t ) + Ayx{t -h 1 ) + ... + A N x(t - h N ) + AA x(t) + AA x x(t -hy)+... + AA N x(t -h N ) + Bu(t) + Df(t)] = s T (f )(CB) _1 [CA x(t) + CA x x(t -h 1 ) + ... + CA N x(t - h N ) + CBE x(t) + CBE- 1 x(t-h^) + ... + CBE N x(t-h N ) -CB((CBy 1 [CA x(t) + CA 1 x(t-h) + ... + CA N x(t-h N )] -[k \\x(t)\\ + k 1 \\x(t-h 1 )\\ + .., + k N \\x(t-h N ) s(t) -Gx{t)-S s(t) '(Oil. -CBEf(t)] --s'(t)[E x(t) + E 1 x(t-h 1 ) + ... + E N x(t-h N )] - [*b NOI + h \W - h )| + -. + k N \Ht ~ h N ) s(t) e(t) Noll -Gx(t)-S- s(t)\\ ■E/(0] * -K*b - «o - s) Noll IKol + ft - «i ) H* - h )|IKO| +...+(fc N -a N )Ht-ft N )|||s(o|]-(«y-/o)|KO| Since (24), (25) hold, then (27) reduces to: V = s^fXCB)" 1 ^) < -{S-f )\\s{t)\\ < - 7 ||s(f)|| where Hence we can evaluate that y(f)<-?; /7 = <?-/ >0 ^in (CB) t#(0 (27) (28) (29) (30) The last inequality (30) is known to prove the finite-time convergence of system (1), (4) towards the sliding surface s(t)=0 (Utkin, 1977), (Perruquetti & Barbot, 2002). Therefore, Theorem 2 is proved. 4.3 Numerical examples and simulation In order to demonstrate the usefulness of the proposed control design techniques let us consider the following examples. Example 1: Consider a networked control time-delay system (1), (4) with parameters taking from (Cao et al, 2007): A = [-4 0" -1 -3 ,A 1 = -1.5 -1 -0.5 (31) Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 171 AA = 0.5sin(£)A ,AA 1 = 05cos(t)A 1 ,/ = 0.3sin(£) The LMI stability and sliding mode existence conditions are computed by MATLAB programming (see Appendix 1) where LMI Control Toolbox is used. The computational results are following: -0.1811 0.1811" 0.3189 -0.3189 AOhat = -1.0866 1.0866 1.9134 -1.9134 : Alhat = Gl = [ 0.9567 1.2933] ; AOtil = eigAOhat = lhs = 0.0000 -3.0000 : eig Alhat : -3.0000 -1.5000 0.0000 -4.5000 0.0000 -0.5000 -1.8137 0.0020 -0.1392 0.1392 0.0020 -1.7813 0.1382 -0.1382 -0.1392 0.1382 -1.7364 0.0010 0.1392 -0.1382 0.0010 -1.7202 ; eigsLHS : ; eigAOtil = -2.0448 -1.7952 -1.7274 -1.4843 -3.0000 -4.5000 P = Rl = BTP 0.6308 -0.0782 . _ 0.3660 -0.0782 0.3891 J ' Glg " [ 0.6539 1.7364 -0.0010] T 1.7202 ; eigRl = -0.0010 1.7202 J & 1.7365 [ 1.1052 0.6217] ; BTPB = 3.4538 invBTPB = 0.2895; normGl = 1.6087 k0= 2.1087; kl=0.5; 8 > 0.3; H< 0; The networked control time-delay system is robustly asymptotically delay-independent stable. Example 2: Consider a time-delay system (1), (4) with parameters: "-1 0.7" ,A 1 = "0.1 0.1" , A 2 = "0.2 " ,B = "1" [o.3 1 L u 0.2 L u 0.1 J 1 hy = 0.1 , h 2 = 0.2 (32) AA n 0.2sin(f) O.lsin(f) ,AAt O.lcos(f) 0.2cos(f) Matching condition for external disturbances is given by: D = BE-- ,AA 0.2cos(f) O.lcos(f) 0.2cost; /(f) = 0.2 cos f The LMI stability and sliding mode existence conditions are computed by MATLAB programming (see Appendix 2) where LMI Control Toolbox is used. The computational results are following: 172 Robust Control, Theory and Applications AOhat = -0.3947 -0.0911' 0.9053 0.2089 ; Alhat = -0.0304 -0.0304' 0.0696 0.0696 ;A2hat = 0.0607 -0.1393 -0.0304 0.0696 Geq=[ 0.6964 0.3036]; G =[ -4.5759 12.7902] AOtil = 4.1812 -12.8812 5.4812 -12.5812 lhs = eigAOhat : -0.7085 -0.5711 -0.0085 0.0084 -0.0085 0.0084 0.0169 -0.1858 0.0000 0.5711 -0.0085 0.8257 0.0084 ■1.0414 ■0.2855 0.0167 ; eigAOtil eig Alhat ; -4.2000 + 0.60001' -4.2000 - 0.60001 0.0393 eigA2hat : -0.0085 0.0169 0.0084 -0.0167 P= -0.0085 0.0084 2.0633 0.778r 0.7781 0.4592 : eigP= -0.2855 -1.1000 0.1438^ 2.3787 -1.0414 -0.2855 :R1= -0.0085 0.0084 -0.2855 -1.1000 1.0414 0.2855 0.2855 1.1000 ; eigsLHS = ;R2= 0.0000 0.1304 -1.3581 -1.3578 -1.3412 -0.7848 -0.7837 -0.1916 1.0414 0.2855 0.2855 1.1000 eigRl 0.7837' 1.3578 : eigR2 = 0.7837' 1.3578 BTP= [ 2.8414 1.2373] ; BTPB = 4.0788 invBTPB= 0.2452; normG = 13.5841 a = 0.2; a x = 0.2; a 2 = 0.2 ; d = max|D| = 0.2 ; f Q = max|/(f)| = 0.2828 ; k0=13.7841; kl=0.2; k2=0.2; 8 > 0.2828; H< 0; Thus, we have designed all the parameters of the combined sliding mode controller. Aircraft control design example 3: Consider the lateral-directional control design of the DC- 8 aircraft in a cruise-flight configuration for M = 0.84, h = 33.000ft, and V = 825ft/ s with nominal parameters taken from (Schmidt, 1998): -0.228 2.148 -0.021 0.0 -1.0 -0.0869 0.0 0.0390 0.335 -4.424 -1.184 0.0 0.0 0.0 1.0 0.0 -1.169 0.065" 0.0223 0.0 \ Sr ] 0.0547 2.120 $a 0.0 0.0 (33) where j3 is the sideslip angle, deg., p is the roll rate, deg/s, <j> is the bank angle, deg., r is the yaw rate, deg/s, S r is the rudder control, S a is the aileron control. However, some small transient time-delay effect in this equation may occur because of influence of sideslip on aerodynamics flow and flexibility effects of aerodynamic airframe and surfaces in lateral- directional couplings and directional-lateral couplings. The gain constants of gyro, rate gyro and actuators are included in to lateral directional equation of motion. Therefore, it is assumed that lateral direction motion of equation contains some delay effect and perturbed parameters as follows: Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 173 -0.002 0.0 ' 0.0 0.004 0.034 -0.442 0.0 0.0 (34) AA = 0.1 Aq sin(f) , AA 1 = 0.1 A 1 cos(f) ,D = I±;f = 0.2sin(f) ; \ = 0.01 - 0.04s The LMI stability and sliding mode existence conditions are computed by MATLAB programming (see Appendix 3) where LMI Control Toolbox is used. The computational results are following: AOhat = G = "-0.0191 -0.0008 0.0000 0.0007" "-0.0000 0.0000 -0.0000 0.0001 -1.0042 -0.0434 0.0003 0.0390 0.0006 0.0000 -0.0000 -0.0000 ; Alhat = -0.0000 0.0003 -0.0000 0.0040 0.0000 -0.0000 0.0000 -0.0000 1.0000 "-0.8539 0.0163 0.0262 0" 0.0220 -0.0001 0.4710 ;Gl = "-0.5925 0.0890 0.1207 0.0501" 0.0689 -0.0086 0.3452 0.0485 AOtil = eigAlhat= 1.0e-003* P = Rl = -0.7162 0.1038 0.1187 0.0561 -0.9910 -0.0454 -0.0024 0.0379 -0.1130 0.0134 -0.7384 -0.1056 1.0000 eigAOtil = [-0.5+0.0821 -0.5-0.082i -0.3 -0.2] eigAOhat = [-0.0621 -0.0004 -0.0000 -0.0000] 0.2577 -0.0000 + 0.00001 -0.0000 - 0.00001 72.9293 39.4515 -2.3218 24.7039^ 39.4515 392.5968 10.8368 -1.4649 -2.3218 10.8368 67.2609 -56.4314 24.7039 -1.4649 -56.4314 390.7773 52.5926 29.5452 0.3864 2.5670 29.5452 62.3324 3.6228 -0.4852 0.3864 3.6228 48.3292 -32.7030 •0.4852 -32.7030 61.2548 -84.5015 -36.7711 6.6350 -31.9983 -0.1819 25.5383 142.4423-118.0289 2.5670 BTP = ; eigP= [57.3353 66.3033 397.7102 402.2156] ; eigRl = [21.3032 27.3683 86.9363 88.9010] BTPB = 98.3252 8.5737 ' 8.5737 301.9658 ; invBTPB = 0.0102 -0.0003 -0.0003' 0.0033 174 Robust Control, Theory and Applications morm = 0.85451hs = -41.4566 -29.8705 -0.6169 -2.3564 -0.0008 0.0105 -0.0016 0.1633' -29.8705 -51.6438 -3.8939 0.8712 -0.0078 0.1015 -0.015 1.5728 -0.6169 -3.8939 -38.2778 32.1696 -0.0002 0.0028 -0.0004 0.043 -2.3564 0.8712 32.1696 -51.6081 -0.0002 -0.0038 -0.0008 -0.0078 -0.0002 -52.593 -29.545 -0.3864 -2.567 0.0105 0.1015 0.0028 -0.0002 -29.545 -62.333 -3.6228 0.4852 -0.0016 -0.015 -0.0004 -0.3864 -3.6228 -48.33 32.703 0.1633 1.5728 0.043 -0.0038 -2.567 0.4852 32.703 -61.255 eigsLHS = -88.9592 -86.9820 -78.9778 -75.8961 -27.3686 -21.3494 -16.0275 -11.9344 k0= 1.0545; kl=0.5; 8 > 0.2; H< 0; Thus, we have designed all the parameters of the aircraft control system and the uncertain time-delay system (1), (4) with given nominal (33) and perturbed (34) parameters are simulated by using MATLAB-SIMULINK. The SIMULINK block diagram of the uncertain time-delay system with variable structure contoller (VSC) is given in Fig. 1. Simulation results are given in Fig. 2, 3, 4 and 5. As seen from the last four figures, system time responses to the rudder and aileron pulse functions (0.3 within 3-6 sec) are stabilized very well for example the settling time is about 15-20 seconds while the state time responses of aircraft control action as shown in Fig. 5 are unstable or have poor dynamic characteristics. Notice that, as shown in Fig. 4, control action contains some switching, however it has no high chattering effects because the continuous terms of controller are dominant. Numerical examples and simulation results show the usefulness and effectiveness of the proposed design approach. 5. Robust delay-dependent stabilization In this section, the existence condition of the sliding manifold and delay-dependent stability analysis of perturbed sliding time-delay systems are presented. 5.1 Robust delay-dependent stabilization on the sliding surface In this section the sliding manifold is designed so that on it or in its neighborhood in different from existing methods the perturbed sliding time-delay system (16) is globally asymptotically stable with respect to state coordinates. The stability results are formulated in the following theorem. Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 175 Time-delay system signal 1 Signal BuildeC Fig. 1. SIMULINK block diagram of uncertain time-delay system with VSC 176 Robust Control, Theory and Applications 0.8 0.6 §0.4 ^ 0.2 [beta p ) phi r] i i i ...A ! ! i / ! /■ M I " i i 10 15 20 25 30 35 40 45 s 50 Fig. 2. States' time responses with control [s-| s 2 ] i i i I i i 1 \\[yY \ \ \ ['■/'■■ ■ \ V / \ . . i i i I i i I Fig. 3. Sliding functions 10 15 20 25 30 35 40 45 s 50 [u R u fl ] i 1 1 Ufl /y u * \ ^"^Jtaj&UHJ HtaOtfflttHfl: iOiaiaixitai)t*aauaatiaaatiaainH x i i i 0.G 0.4 0.2 ■0.2 ■0.4 5 10 15 20 25 30 35 40 45 s 50 Fig. 4. Control functions Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 177 1.5 1 [beta n phi r] ! I ' 0.5 riR pP/V r ■ f^" i i I i 10 15 35 40 45 s 50 Fig. 5. States' time responses without control Theorem 3: Suppose that Assumption 1 holds. Then the transformed multivariable sliding time-delay system (16) with matched parameter perturbations and external disturbances driven by combined controller (4) and restricted to the sliding surface s(t)=0 is robustly globally asymptotically delay-dependent stable with respect to the state variables, if the following modified LMI conditions and parameter requirements are satisfied: H v H-- PA t A -PA{ -PA,A N -PA N A ~PA N A, .. • -pK h l h i 1 h N 1 s -T 7 :0 (35) where H 11 =(A 0+ A 1 +...+A N ) T P + P(A 0+ A 1 +...+A N ) + h 1 (S 1 +RO+...+h N (S N +R N )+T 1 +^ CB = B 1 PB>0 k -a§)k x -a 1 ',...k A (36) (37) 178 Robust Control, Theory and Applications S^fo (38) where P,R 1 ,...R N are some symmetric positive definite matrices which are a feasible solution of modified LMI (35) with (36); A = A -BG is a stable matrix. Proof: Let us choose a special augmented Lyapunov-Krasovskii functional as follows: N t V = x T (t)Px(t) + J] j* j* x T (p)R i x(p)dp d6 i=l-hi t+e N t n t +1 J J x T (p)S i x(p)dpd0^ J x T {0)T i x{e)dO (39) The introduced special augmented functional (39) involves three particular terms: first term Vi is standard Lyapunov function, second and third are non-standard terms, namely V2 and V3 are similar, except for the length integration horizon [t-h, t] for V2 and [t+6-h, t] for V3, respectively. This functional is different from existing ones. The time-derivative of (39) along the perturbed time-delay system (16) can be calculated as: V = x T (t)[(A 0+ A 1+ ... + A N ) T P + P(A 0+ A 1+ ... + A N ) + h 1 (S 1 +R 1 ) + ... + h N (S N +R N ) + T 1 +... + T N x(t)] t t t -2x T (t)PA t A j" x(0)dO-2x T (t)PAl j" x(0-h 1 )d0-...-2x T (t)PA 1 A N J x(0-h N )dO t-hi t-hi t-hi t t t -...-2x T (t)PA N A J x(6)d6-2x T (t)PA N A 1 J x(0-h 1 )d0-...-2x T (t)PAl j" x{6-h N )dO t-h N t-h N t-h N t t -\ J x T (0)R 1 x(0)d0-...-h N J x T (6 )R N x(0 )d0 t-h, t -\ J x T (6-\ )S 1 x(6-\ )d0-...-h N j x T (6>-h N )S N x(0-h N )d0 t-K t-h N -x T (t-h l )T 1 x(t-h 1 )-x T (t-h N )T N x(t-h N ) (40) +2x T (f)PzlA 1 x(f-/z 1 ) + ... + 2x T (f)PzlA N x(f-/z ]V )-2x T (f)PB[/c |x(f)| + /c 1 |x(f-/z 1 ) + .., + ^|x(f-/z N )|]^-2x T (0PB^^ + x T (f)PD/(0 Since for some h>0 Noldus inequality holds: t \ J x T {6)R 1 x{e)de> t-h t f j x(0 )d0 t-H-y h N J x T (6>-h N )S N x(6>-h N )d6>> t-h N R 1 j x(0 )d0 t-H-y t J x(0-h N )d0 (41) j x(0-h N )d0 Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 179 and x T (t)PB = s T (t) then (40) becomes as: V<x T (t)[(A +A 1 +... + A N ) T P + P(A +A 1 +... + A N ) + h 1 (S 1 +R 1 ) + ... + h N (S N t t -2x T (0PA 1 A J x{0)d0-2x T {t)PA\ J x(0-h 1 )d0 _ _ t _ _ t -...-2x T (t)PA 1 A N J x(0 -h N )d0 - ...- 2x T (t)PA N A J x(0)d0 t-hy t-h N t t 1 -2x T (t)PA N A 1 J x(0-h 1 )de-...-2x T (t)PAl J x(0-h N )d0 t-h N t-h N J x{0)dO J x(0-h N )d0 J x(6>)^ J x(0-h 1 )d0 J x(<9)d<9 J x(0-h 1 )d0 J x(0)d0 t-hy | x(0-h N )d0 -x T (t-h 1 )T 1 x(t-h 1 )-x T (t-h N )T N x(t-h N ) +2x J \t)PBE x x(t -h 1 ) + ... + 2x T (t)PBE N x(t -h N )- 2x T (t)PB[k \\x(t)\\ + k x \\x(t - h x )\ + .., + ^ N |^-/z N )|]^-2x T (0PB^^ + x T (0PBE/(0 <x r (0[(A +A 1+ ... + A N ) r P + P(A +^^ t t t -2x T (t)PA 1 A J x(6)d6-2x T (t)PAl J x(0-h 1 )d0 - ...-2x T (t)PA t A N J x(0-h N )d0 t-hy t-hy t-hy _ _ t _ _ t t -...-2x T (t)PA N A J x(0)d0-2x T (t)PA N A 1 J x(0-h 1 )d0 - ...-2x T (t)PA^ J x(0-h N )d0 t-h N t-h N t-h N 1_ J x(6>)^ T t t T t - R x J x(0)d0 1 J x{0)d0 R N J x(0)d0 L*-^ J J-h N J-h N -i T i t T r )d0 s a J x(0-h 1 )d0 1 h N J x(0-h N )d0 $N t-hy J-h„ j x(0-h N )d0 -x T (t-h 1 )T 1 x(t-h 1 )-x T (t-h N )T N x(t-h N ) + 2s T (t)E 1 x(t-h 1 ) + ... + 2s(t)E N x(t-h N ^ -2s T (t) [k \\x(t)\\ + k x \\x(t - hy )|| + ... . + k N \\x(t - h h ^ 2Ss T (t)^ + s T (t)Ef(t) \W)\\ \W)\\ x(t) J x(0)d0 j x{0-h r )d0 j x(0-h N )d0 j x(0)d0 J x{0 -h r )d0 j x(0-h N )d0 x{t-h r ) x(t-h N ) t-h, t-h. t-h N t-h N 180 Robust Control, Theory and Applications x(t) Hi, -PA.A, -PAl - -PA-^A N —PA N A Q —PAjyA-^ -^ t | x(6)d6 t-h x * -J-R, hi t | x{e-h x )&0 * -— S] ha t-h x \ x(9-h N )d9 * 1 -t-Rn h N t-h x t * | x(0)dO * t | x{e-\)dG * t-h N -T 2 o '-. t | x(0-h N )dO t-h N -v x(t-h t ) x(t-h N ) -K*o -a )|x(0||s(0| + (^ -a 1 )|^-fi 1 )||sW| + ... + (fc N - -a N )\\x(t -h n)\\\W)\\] -(S -fo)\W)\\ (42) Since (35)-(38) hold, then (42) reduces to: V<z T (0Hz(0< (43) Therefore, we can conclude that the perturbed time-delay system (16), (4) is robustly globally asymptotically delay-dependent stable. Theorem 3 is proved. Special case: Single state-delayed systems: For single state-delayed systems that are frequently encountered in control applications and testing examples equation of motion and control algorithm can be easily found from (1), (4), (16) letting N=l. Therefore, the modified LMI delay-dependent stability conditions for which are significantly reduced and can be summarized in the following Corollary. Corollary 1: Suppose that Assumption 1 holds. Then the transformed single-delayed sliding system (16) with matched parameter perturbations and external disturbances driven by combined controller (4) for which N=l and restricted by sliding surface s(t)=0 is robustly globally asymptotically delay-dependent stable with respect to the state variables, if the following LMI conditions and parameter requirements are satisfied: H-- ( A)+ A 1 fP + P(A 0+ A 1 ) 2 +h l (S 1 +R L ) + T 1 -(PA,A Q ) T -(PA\f o Ri o K o o o <0 (44) Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 181 CB = B T PB > (45) k =a ;k 1 =a 1 ; (46) S>f (47) Proof: The corollary follows from the proof of the Theorem 3 letting N=l. 5.2 Existence conditions The final step of the control design is the derivation of the sliding mode existence conditions or the reaching conditions for the perturbed time-delay system states to the sliding manifold in finite time. These results are summarized in the following theorem. Theorem 4: Suppose that Assumption 1 holds. Then the perturbed multivariable time- delay system (1) states with matched parameter uncertainties and external disturbances driven by controller (4) converge to the siding surface s(t)=0 in finite time, if the following conditions are satisfied: k =a + g)k x = a 1 ;... / k N = a N ; (48) S>f (49) Proof: Let us choose a modified Lyapunov function candidate as: V=h T (t)(CB)-'s(t) (50) The time-derivative of (50) along the state trajectories of time-delay system (1), (4) can be calculated as follows: V = s T (0(CB)" 1 s(0 = s T (0(CB)" 1 Cx(0 = s T (f)(CB)" 1 C[A x(0 + A x x{t - \) + ... + A N x(t -h N ) + AA x(t) + AA x x(i - h 1 )+... + AA N x(t -h N ) + Bu(t) + Df(t)] = s T (0(CB) -1 [CA x(f) + CA 1 x(f-/z 1 ) + ... + CA N x(f-/z N ) + CBE x(f) + CBE 1 x(f-/z 1 ) + ... + CBE N x(t-h N )-CB((CB)~ 1 [CA x(t) + CA 1 x(t-h) + ... + CA N x(t-h N )] - [*b HOII + *i lk(* - ^i)ll + «.. + ^ lk(* - ^)ll] FTSir - Gx (*)-^ FTSii I + c BE /(*)] ( 51 ) KOI IKOIU = s T (t)[E x(t) + Etfit -h 1 ) + ... + E N x(t - h N) - [*o \\ X (t)\\ + k \\ X (t - h,)\\ + .... + k N \\ X (t - MOirSn Gx(f) " ^ftt^ + E W\ KOI K0| ^ -[(^o - «o - ^)||^(^)||IN(^)|| + (^a - «i)||^(^ - ^a)||||^(0|| + ... + (^-^)||x(t-/z^)||||s(0||]-(^-/ )N0|| Since (48), (49) hold, then (51) reduces to: V = s T (0(CB)- a s(0 < -{5 - f )\\s(t)\\ < -r,\\s(t)\\ (52) 182 Robust Control, Theory and Applications where Hence we can evaluate that rj = S-f >0 (53) V(t): Anin (CB)" 1 V(t) (54) The last inequality (54) is known to prove the finite-time convergence of system (1),(4) towards the sliding surface s(t)=0 (Utkin, 1977), (Perruquetti & Barbot, 2002). Therefore, Theorem 4 is proved. 5.3. Numerical examples In order to demonstrate the usefulness of the proposed control design techniques let us consider the following examples. Example 4: Consider a time-delay system (1),(4) with parameters taken from (Li & De Carlo, 2003): 2 1 " 1.75 0.25 0.8 ;A 1 = -1 1 -1 " "0" -0.1 0.25 0.2 ;B = -0.2 4 5 1 AA = 0.2sin(f)A / AA 1 = 02cos(t)A 1 ,/ = 0.3sin(f) The LMI delay-dependent stability and sliding mode existence conditions are computed by MATLAB programming (see Appendix 4) where LMI Control Toolbox is used. The computational results are following: [ 1.2573 2.5652 1.0000] AOhat = eigAOhat : Geq 2.0000 1.0000 " 1.7500 0.2500 0.8000 -7.0038 -0.6413 -3.3095 : Alhat = -1.0000 -0.1000 0.2500 0.2000 1.5139 -0.6413 -0.5130 -0.5298 + 0.53831 -0.5298 - 0.53831 0.0000 ; eigAlhat = [ -0.2630 -0.0000 -1.0000] AOtil = G= [3.3240 10.7583 3.2405]; Geq =[ 1.2573 2.5652 1.0000] ; eigAOtil = 2.0000 1.0000 1.7500 0.2500 0.8000 -10.3278 -11.3996 -6.5500 -2.7000 -0.8000 + 0.50001 -0.8000 - 0.50001 P= 1.0e+008* 1.1943 -1.1651 0.1562 -1.1651 4.1745 0.3597 0.1562 0.3597 0.1248 ;R1= 1.0e+008* 1.9320 0.2397 0.8740 0.2397 1.0386 0.2831 0.8740 0.2831 0.4341 Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 183 Sl= 1.0e+008* lhs = 1.0e+008 * 0.8783 0.1869 0.2951 0.1869 1.0708 0.2699 0.2951 0.2699 0.1587 ;T1= 1.0e+007* -1.1632 0.4424 -0.1828 0.1743 -0.1030 0.1181 -0.4064-0.1030 -0.0824 0.4424 -1.6209 -0.1855 0.5480 0.2138 0.2098 0.3889 0.2138 0.1711 -0.1828 -0.1855 -0.0903 0.0445 0.0026 0.0215 -0.0142 0.0026 0.0021 0.1743 0.5480 0.0445 -1.9320 -0.2397 -0.8740 -0.1030 0.2138 0.0026 -0.2397 -1.0386 -0.2831 0.1181 0.2098 0.0215 -0.8740 -0.2831 -0.4341 -0.4064 0.3889 -0.0142 -0.8783 -0.1869 -0.2951 -0.1030 0.2138 0.0026 -0.1869 -1.0708 -0.2699 -0.0824 0.1711 0.0021 -0.2951 -0.2699 -0.1587 -0.2362 0.0730 0.0730 -0.7576 -0.0726 -0.1159 2.3624 -0.7303 0.7264 -0.7303 7.575S 1.1589 0.7264 1.1589 0.4838 -0.0726 -0.1159 -0.0484 maxhl = 1; eigsLHS = 1.0e+008 ' -2.8124 -2.0728 -1.0975 -0.9561 -0.8271 -0.7829 -0.5962 -0.2593 -0.0216 -0.0034 -0.0000 -0.0000 ; NormP = 4.5946e+008 G = [ 3.3240 10.7583 3.2405] ; NormG = 11.7171 invBtPB= 8.0109e-008; BtP = 1.0e+007*[ 1.5622 3.5970 1.2483] eigP = 1.0e+008 * eigSl = 1.0e+008 * 0.0162 0.8828 4.5946 "0.0159" 0.7770 1.3149 ; eigRl = 1.0e+008 ' ; eigTl = 1.0e+007 * 0.0070 0.9811 2.4167 0.0000^ 2.5930 7.8290 184 Robust Control, Theory and Applications k0= 11.9171; kl=0.2; 8 > 0.3; H<0 Considered time-delay system is delay-dependently robustly asymptotically stable for all constant delays h < 1 . Example 5: Now, let us consider a networked control time-delay system (1), (4) with parameters taken from (Cao et al., 2007): "-4 0" ,A 1 = -1.5 ,B = ~2 [-1 -3 L _1 -0.5 L 2 J A = AA =0.5sin(t)A ,AA 1 =05cos(t)A 1 ,f = 0.3sin(£) The LMI delay-dependent stability and sliding mode existence conditions are computed by MATLAB programming (see Appendix 5) where LMI Control Toolbox is used. The computational results are following: maxhl = 2.0000; Geq = [ 0.4762 0.0238] AOhat = -0.1429 0.1429 2.8571 -2.8571 ; Alhat = -0.0238 0.0238 0.4762 -0.4762 eigAOhat : -0.0000 -3.0000 AOtil = -4.1429 -0.0571 -1.1429 -3.0571 ; eigAlhat ; eigAOtil = -0.0000 -0.5000 -4.2000 -3.0000 P = 1.0e+004* 5.7534 -0.1805 -0.1805 0.4592 ; Rl = 1.0e+004 * 8.4457 -0.2800 -0.2800 0.6883 SI = 1.0e+004 * 7.7987 0.2729 0.2729 0.1307 ; Tl = 1.0e+004 * lhs = 1.0e+004* 6.7803 0.3390 0.3390 0.0170 -8.4351 1.2170 -0.6689 0.6689 -0.1115 0.1115 1.2170 -1.5779 0.6689 -0.6689 0.6689 -4.2228 0.6689 -0.6689 0.1400 -0.1115 0.1115 0.1115 -0.1115 -0.6689 0.1115 0.1400 -0.3442 0.1115 -3.8994 -0.1364 -0.1364 -0.0653 -6.7803 -0.3390 -0.3390 -0.0170 Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 185 eigsLHS = 1.0e+004 * -8.8561' -6.7973 -4.1971 -3.9040 -1.4904 -0.0971 -0.0000 -0.0000 NormP = 5.7595e+004; G = [ 2.0000 0.1000] NormG = 2.0025; invBtPB = 4.2724e-006; BtP = 1.0e+005 * [1.1146 0.0557] eigsP = 1.0e+004 * 0.4530 5.7595 ; eigsRl = 1.0e+004 * 0.6782 8.4558 eigsSl = 1.0e+004 * 0.1210" 7.8084 eigsTl = 1.0e+004* 0.0000" 6.7973 k0= 2.5025; kl=0.5; 8 > 0.3; H< The networked control time-delay system is robustly asymptotically delay-dependent stable for all constant time-delays h < 2.0000 . Thus, we have designed all the parameters of the combined sliding mode controller. Numerical examples show the usefulness of the proposed design approach. 6. Conclusion The problem of the sliding mode control design for matched uncertain multi-input systems with several fixed state delays by using of LMI approach has been considered. A new combined sliding mode controller has been proposed and designed for the stabilization of uncertain time-delay systems with matched parameter perturbations and external disturbances. Delay-independent and delay-dependent global stability and sliding mode existence conditions have been derived by using Lyapunov-Krasovskii functional method and formulated in terms of linear matrix inequality techniques. The allowable upper bounds on the time-delay are determined from the LMI stability conditions. These bounds are independent in different from existing ones of the parameter uncertainties and external disturbances. Five numerical examples and simulation results with aircraft control application have illustrated the usefulness of the proposed design approach. The obtained results of this work are presented in (Jafarov, 2008), (Jafarov, 2009). 7. Appendices A1 clear; clc; 186 Robust Control, Theory and Applications A0=[-4 0;-l -3]; Al=[-1.5 0;-1 -0.5]; B=[2; 2]; setlmis([]) P =lmivar(l,[2 1]); Rl=lmivar(l,[2 1]); Geq=inv(B'*P*B)*B'*P A0hat=A0-B*G*A0 Alhat=Al-B*G*Al G= place(A0hat,B,[-4.5 -3]) A0til=A0hat-B*Gl eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) ii = 1; lmiterm([-l 1 1 P],ii,ii) lmiterm([-2 1 1 Rl],ii,ii) lmiterm([4 1 1 P],l,A0tir;s') lmiterm([4 1 1 Rl],ii,ii) lmiterm([4 2 2 Rl],-ii,ii) lmiterm([4 1 2 P],l,Alhat) LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs P eigP=eig(P) Rl eigRl=eig(Rl) eigsLHS=eig(lhs) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) % recalculate Geq=inv(B'*P*B)*B , *P A0hat=A0-B*G*A0 Alhat=Al-B*G*Al G= place(A0hat,B,[-4.5 -3]) A0til=A0hat-B*Gl eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 87 ii = 1; setlmis([]) P =lmivar(l,[2 1]); Rl=lmivar(l,[2 1]); R2=lmivar(l,[2 1]); lmiterm([-l 1 1 P],ii,ii) lmiterm([-2 1 1 Rl],ii,ii) lmiterm([4 1 1 PJ^AOtil'/s') lmiterm([4 1 1 Rl],ii,ii) lmiterm([4 2 2 Rl],-ii,ii) lmiterm([4 1 2 P],l,Alhat) LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs P eigP=eig(P) Rl eigRl=eig(Rl) eigsLHS=eig(lhs) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) normGl = norm(Gl) A2 clear; clc; A0=[-1 0.7; 0.31]; Al=[-0.1 0.1; 0.2]; A2=[0.2 0;0 0.1]; B=[l; 1] setlmis([]) P=lmivar(l,[21]); Rl=lmivar(l,[2 1]); R2=lmivar(l,[2 1]); Geq=inv(B , *P*B)*B'*P A0hat=A0-B*G*A0 Alhat=Al-B*G*Al A2hat=A2-B*G*A2 G= place(A0hat,B,[-4.2-.6i -4.2+.6i]) A0til=A0hat-B*Gl 188 Robust Control, Theory and Applications eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) eigA2hat=eig(A2hat) ii = l; lmiterm([-l 1 1 P],ii,ii) lmiterm([-2 1 1 Rl],ii,ii) lmiterm([-3 1 1 R2],ii,ii) lmiterm([4 1 1 P],l,A0tir;s') lmiterm([4 1 1 Rl],ii,ii) lmiterm([4 1 1 R2],ii,ii) lmiterm([4 2 2 Rl],-ii,ii) lmiterm([4 1 2 P],l,Alhat) lmiterm([4 1 3 P],l,A2hat) lmiterm([4 3 3 R2],-ii,ii) LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); R2=dec2mat(LMISYS,xopt,R2); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs eigsLHS=eig(lhs) P eigP=eig(P) Rl R2 eigRl=eig(Rl) eigR2=eig(R2) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) % recalculate Geq=inv(B'*P*B)*B , *P A0hat=A0-B*G*A0 Alhat=Al-B*G*Al A2hat=A2-B*G*A2 G= place(A0hat,B,[-4.2-.6i -4.2+.61]) A0til=A0hat-B*Gl eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) eigA2hat=eig(A2hat) ii = 1; Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 89 setlmis([]) P =lmivar(l,[2 1]); Rl=lmivar(l,[2 1]); R2=lmivar(l,[2 1]); lmiterm([-l 1 1 P],ii,ii) lmiterm([-2 1 1 Rl],ii,ii) lmiterm([-3 1 1 R2],ii,ii) lmiterm([4 1 1 P],l,A0tir;s') lmiterm([4 1 1 Rl],ii,ii) lmiterm([4 1 1 R2],ii,ii) lmiterm([4 2 2 Rl],-ii,ii) lmiterm([4 1 2 P],l,Alhat) lmiterm([4 1 3 P],l,A2hat) lmiterm([4 3 3 R2],-ii,ii) LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); R2=dec2mat(LMISYS,xopt,R2); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs eigsLHS=eig(lhs) P eigP=eig(P) Rl R2 eigRl=eig(Rl) eigR2=eig(R2) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) normGl = norm(Gl) A3 clear; clc; A0= [-0.228 2.148 -0.021 0; -1 -0.0869 0.039; 0.335 -4.424 -1.184 0; 10]; A1=[0 -0.002 0; 0.004; 0.034-0.442 0; 0]; B =[-1.169 0.065; 0.0223 0; 0.0547 2.120; 0]; setlmis([]) P =lmivar(l,[4 1]); Rl=lmivar(l,[4 1]); G=inv(B'*P*B)*B , *P A0hat=A0-B*G*A0 190 Robust Control, Theory and Applications Alhat=Al-B*G*Al Gl= place(A0hat,B,[-.5+.082i -.5-.0821 -.2 -.3]) A0til=A0hat~B*Gl eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) %break ii = l; lmiterm([-l 1 1 P],ii,ii) lmiterm([-2 1 1 Rl],ii,ii) lmiterm([4 1 1 P],l,A0tir;s') lmiterm([4 1 1 Rl],ii,ii) lmiterm([4 2 2 Rl],-ii,ii) lmiterm([4 1 2 P],l,Alhat) LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs P eigP=eig(P) Rl eigRl=eig(Rl) eigsLHS=eig(lhs) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) gnorm=norm(G) A4 clear; clc; A0=[2 1; 1.75 0.25 0.8; -1 1] Al=[-1 0; -0.1 0.25 0.2; -0.2 4 5] B =[0;0;1] %break hl=1.0; setlmis([]); P=lmivar(l,[3 1]); Geq=inv(B'*P*B)*B'*P A0hat=A0-B*Geq*A0 Alhat=Al-B*Geq*Al eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 91 DesPol = [-2.7 -.8+.5i -.8-.5i]; G= place(AOhat,B,DesPol) A0til=A0hat-B*G eigA0til=eig(A0til) Rl=lmivar(l,[3 1]); Sl=lmivar(l,[3 1]); Tl=lmivar(l,[3 1]); lmiterm([-l 1 1 P],l,l); lmiterm([-l 2 2 Rl],l,l); lmiterm([-2 1 1 Sl],l,l); lmiterm([-3 1 1 Tl],l,l); lmiterm([4 1 1 P] / (AOtil+Alhat) , / l / , s'); lmiterm([4 1 1 Sl],hl,l); lmiterm([4 1 1 Rl],hl,l); lmiterm([4 1 1 Tl],l,l); lmiterm([4 1 2 P],-l,Alhat*AOhat); lmiterm([4 1 3 P],-l,Alhat*Alhat); lmiterm([4 2 2 Rl],-l/hl,l); lmiterm([4 3 3 Sl],-l/hl,l); lmiterm([4 4 4 Tl],-l,l); LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); Sl=dec2mat(LMISYS,xopt,Sl); Tl=dec2mat(LMISYS,xopt,Tl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs,hl,P,Rl,Sl,Tl eigsLHS=eig(lhs) % repeat clc; Geq=inv(B , *P*B)*B , *P A0hat=A0-B*Geq*A0 Alhat=Al-B*Geq*Al eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) G= place(AOhat,B,DesPol) A0til=A0hat-B*G eigA0til=eig(A0til) setlmis([]); P=lmivar(l,[3 1]); Rl=lmivar(l,[3 1]); Sl=lmivar(l,[3 1]); Tl=lmivar(l,[3 1]); 192 Robust Control, Theory and Applications lmiterm([-l 1 1 P],l,l); lmiterm([-l 2 2 Rl],l,l); lmiterm([-2 1 1 Sl],l,l); lmiterm([-3 1 1 Tl],l,l); lmiterm([4 1 1 P^AOtil+Alhat^l/s'); lmiterm([4 1 1 Sl],hl,l); lmiterm([4 1 1 Rl],hl,l); lmiterm([4 1 1 Tl],l,l); lmiterm([4 1 2 P],-l,Alhat*AOhat); lmiterm([4 1 3 P],-l,Alhat*Alhat); lmiterm([4 2 2 Rl],-l/hl,l); lmiterm([4 3 3 Sl],-l/hl,l); lmiterm([4 4 4 Tl],-l,l); LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); Sl=dec2mat(LMISYS,xopt,Sl); Tl=dec2mat(LMISYS,xopt,Tl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs,hl,P,Rl,Sl,Tl eigLHS=eig(lhs) NormP=norm(P) G NormG = norm(G) invBtPB=inv(B'*P*B) BtP=B'*P eigP=eig(P) eigRl=eig(Rl) eigSl=eig(Sl) eigTl=eig(Tl) A5 clear; clc; A0=[-4 0;-l -3]; Al=[-1.5 0;-1 -0.5]; B=[2; 2]; hl=2.0000; setlmis([]); P=lmivar(l,[2 1]); Geq=inv(B'*P*B)*B , *P A0hat=A0-B*Geq*A0 Alhat=Al-B*Geq*Al eigA0hat=eig(A0hat) Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 93 eigAlhat=eig(Alhat) % DesPol = [~.8+.5i -.8-.5i]; G= place (AOhat,B,DesPol); avec = [2 0.1]; G = avec; A0til=A0hat-B*Gl eigA0til=eig(A0til) Rl=lmivar(l,[2 1]); Sl=lmivar(l,[2 1]); Tl=lmivar(l,[21]); lmiterm([-l 1 1 P],l,l); lmiterm([-l 2 2 Rl],l,l); lmiterm([-2 1 1 Sl],l/L); lmiterm([-3 1 1 Tl],l,l); lmiterm([4 1 1 P^AOtil+Alhat^l/s'); lmiterm([4 1 1 Sl],hl,l); lmiterm([4 1 1 Rl],hl,l); lmiterm([4 1 1 Tl],l,l); lmiterm([4 1 2 P],-l,Alhat*A0hat); lmiterm([4 1 3 P],-l,Alhat*Alhat); lmiterm([4 2 2 Rl],-l/hl,l); lmiterm([4 3 3 Sl],-l/hl,l); lmiterm([4 4 4 Tl],-l,l); LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); Sl=dec2mat(LMISYS,xopt,Sl); Tl=dec2mat(LMISYS,xopt,Tl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs,hl,P,Rl,Sl,Tl eigsLHS=eig(lhs) % repeat Geq=inv(B , *P*B)*B , *P A0hat=A0-B*Geq*A0 Alhat=Al-B*Geq*Al eigA0hat=eig(A0hat) eigAlhat=eig(Alhat) G = avec; A0til=A0hat-B*G eigA0til=eig(A0til) setlmis([]); P=lmivar(l,[21]); Rl=lmivar(l,[2 1]); Sl=lmivar(l,[2 1]); 194 Robust Control, Theory and Applications Tl=lmivar(l,[21]); lmiterm([-l 1 1 P],l,l); lmiterm([-l 2 2 Rl],l,l); lmiterm([-2 1 1 Sl],l,l); lmiterm([-3 1 1 Tl],l,l); lmiterm([4 1 1 P],(AOtil+Alhat) , ,l, , s'); lmiterm([4 1 1 Sl],hl,l); lmiterm([4 1 1 Rl],hl,l); lmiterm([4 1 1 Tl],l,l); lmiterm([4 1 2 P],-l,Alhat*AOhat); lmiterm([4 1 3 P],-l,Alhat*Alhat); lmiterm([4 2 2 Rl],-l/hl,l); lmiterm([4 3 3 Sl],-l/hl,l); lmiterm([4 4 4 Tl],-l,l); LMISYS=getlmis; [copt,xop t] =f easp (LMISYS); P=dec2mat(LMISYS,xopt,P); Rl=dec2mat(LMISYS,xopt,Rl); Sl=dec2mat(LMISYS,xopt,Sl); Tl=dec2mat(LMISYS,xopt,Tl); evlmi=evallmi(LMISYS,xopt); [lhs,rhs] =showlmi(evlmi,4); lhs,hl,P,Rl,Sl,Tl eigsLHS=eig(lhs) NormP=norm(P) G NormG = norm(G) invBtPB=inv(B'*P*B) BtP=B'*P eigsP=eig(P) eigsRl=eig(Rl) eigsSl=eig(Sl) eigsTl=eig(Tl) 8. References Utkin, V. I. (1977), Variable structure system with sliding modes, IEEE Transactions on Automatic Control, Vol. 22, pp. 212-222. Sabanovic, A.; Fridman, L. & Spurgeon, S. (Editors) (2004). Variable Structure Systems: from Principles to Implementation, The Institution of Electrical Engineering, London. Perruquetti, W. & Barbot, J. P. (2002). Sliding Mode Control in Engineering, Marcel Dekker, New York. Richard J. P. (2003). Time-delay systems: an overview of some recent advances and open problems, Automatica, Vol. 39, pp. 1667-1694. Robust Delay-lndependent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 1 95 Young, K. K. O.; Utkin, V. I. & Ozgiiner, U. (1999). A control engineer's guide to sliding mode control, Transactions on Control Systems Technology, Vol. 7, No. 3, pp. 328-342. Spurgeon, S. K. (1991). 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New results on delay-dependent robust stability of uncertain time delay systems, International Journal of Systems Science, Vol. 41, No. 6, pp. 627-634. Schmidt, L. V. (1998). Introduction to Aircraft Flight Dynamics, AIAA Education Series, Reston, VA. Jafarov, E. M. (2008). Robust delay-dependent stabilization of uncertain time-delay systems by variable structure control, Proceedings of the International IEEE Workshop on Variable Structure Systems VSS'08, pp. 250-255, June 2008, Antalya, Turkey. Jafarov, E. M. (2009). Robust sliding mode control of multivariable time-delay systems, Proceedings of the 11th WSEAS International Conference on Automatic Control, Modelling and Simulation, pp. 430-437, May-June 2009, Istanbul, Turkey. A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System Masanao Obayashi, Norihiro Nakahara, Katsumi Yamada, Takashi Kuremoto, Kunikazu Kobayashi and Liangbing Feng Yamaguchi University Japan 1. Introduction In this chapter, a novel control method using a reinforcement learning (RL) (Sutton and Barto (1998)) with concept of sliding mode control (SMC) (Slotine and Li (1991)) for unknown dynamical system is considered. In designing the control system for unknown dynamical system, there are three approaches. The first one is the conventional model-based controller design, such as optimal control and robust control, each of which is mathematically elegant, however both controller design procedures present a major disadvantage posed by the requirement of the knowledge of the system dynamics to identify and model it. In such cases, it is usually difficult to model the unknown system, especially, the nonlinear dynamical complex system, to make matters worse, almost all real systems are such cases. The second one is the way to use only the soft-computing, such as neural networks, fuzzy systems, evolutionary systems with learning and so on. However, in these cases it is well known that modeling and identification procedures for the dynamics of the given uncertain nonlinear system and controller design procedures often become time consuming iterative approaches during parameter identification and model validation at each step of the iteration, and in addition, the control system designed through such troubles does not guarantee the stability of the system. The last one is the way to use the method combining the above the soft-computing method with the model-based control theory, such as optimal control, sliding mode control (SMC), H^ control and so on. The control systems designed through such above control theories have some advantages, that is, the good nature which its adopted theory has originally, robustness, less required iterative learning number which is useful for fragile system controller design not allowed a lot of iterative procedure. This chapter concerns with the last one, that is, RL system, a kind of soft-computing method, supported with robust control theory, especially SMC for uncertain nonlinear systems. RL has been extensively developed in the computational intelligence and machine learning societies, generally to find optimal control policies for Markovian systems with discrete state and action space. RL-based solutions to the continuous-time optimal control problem have been given in Doya (Doya (2000). The main advantage of using RL for solving optimal 198 Robust Control, Theory and Applications control problems comes from the fact that a number of RL algorithms, e.g. Q-learning (Watkins et al. (1992)) and actor-critic learning (Wang et al. (2002)) and Obayashi et al. (2008)), do not require knowledge or identification/ learning of the system dynamics. On the other hand, remarkable characteristics of SMC method are simplicity of its design method, good robustness and stability for deviation of control conditions. Recently, a few researches as to robust reinforcement learning have been found, e.g., Morimoto et al. (2005) and Wang et al. (2002) which are designed to be robust for external disturbances by introducing the idea of H*, control theory (Zhau et al. (1996)), and our previous work (Obayashi et al. (2009)) is for deviations of the system parameters by introducing the idea of sliding mode control commonly used in model-based control. However, applying reinforcement learning to a real system has a serious problem, that is, many trials are required for learning to design the control system. Firstly we introduce an actor-critic method, a kind of RL, to unite with SMC. Through the computer simulation for an inverted pendulum control without use of the inverted pendulum dynamics, it is clarified the combined method mentioned above enables to learn in less trial of learning than the only actor-critic method and has good robustness (Obayashi et al. (2009a)). In applying the controller design, another problem exists, that is, incomplete observation problem of the state of the system. To solve this problem, some methods have been suggested, that is, the way to use observer theory (Luenberger (1984)), state variable filter theory (Hang (1976), Obayashi et al. 2009b) and both of the theories (Kung and Chen (2005)). Secondly we introduce a robust reinforcement learning system using the concept of SMC, which uses neural network- type structure in an actor/ critic configuration, refer to Fig. 1, to the case of the system state partly available by considering the variable state filter (Hang (1976)). Critic 1 fc fe w P(t) f W *■ JNIoise denerator r/fN i f *a\ r(t) ^ n(t) 1 * s „ . / <f w ^ Environment 1— ► Actor - u(t) W v~ x(t) Fig. 1. The construction of the actor-critic system, (symbols in this figure are reffered to section 2) The rest of this chapter is organized as follows. In Section 2, the conventional actor-critic reinforcement learing system is described. In Section 3, the controlled system, variable filter and sliding mode control are shortly explained. The proposed actor-critic reinforcement learning system with state variable filter using sliding mode control is described in Section 4. Comparison between the proposed system and the conventional system through simulation experiments is executed in Section 5. Finally, the conclusion is given in Section 6. A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 199 2. Actor-critic reinforcement learning system Reinforcement learning (RL, Sutton and Barto (1998)), as experienced learning through trial and error, which is a learning algorithm based on calculation of reward and penalty given through mutual action between the agent and environment, and which is commonly executed in living things. The actor-critic method is one of representative reinforcement learning methods. We adopted it because of its flexibility to deal with both continuous and discrete state-action space environment. The structure of the actor-critic reinforcement learning system is shown in Fig. 1. The actor plays a role of a controller and the critic plays role of an evaluator in control field. Noise plays a part of roles to search the optimal action. 2.1 Structure and learning of critic 2.1.1 Structure of critic The function of the critic is calculation of P(t) : the prediction value of sum of the discounted rewards r(t) that will be gotten over the future. Of course, if the value of P(t) becomes bigger, the performance of the system becomes better. These are shortly explained as follows. The sum of the discounted rewards that will be gotten over the future is defined as V(t) . 00 1=0 where /(0</<l)isa constant parameter called discount rate. Equation (1) is rewritten as V(t) = r(t) + yV(t + l). (2) Here the prediction value of V(t) is defined as P(f) . The prediction error r(t) is expressed as follows, f(t) = f t =r(t) + y P(t + l)-P(t). (3) The parameters of the critic are adjusted to reduce this prediction error r (t ) . In our case the prediction value ~P(t) is calculated as an output of a radial basis function neural network (RBFN) such as, p(0=Z«W), ( 4 ) 7=1 y c j(t) = exp -I(*iW-4) 2 /K-> i=l (5) Here, y c At) :;th node's output of the middle layer of the critic at time t ,co C :\ the weight of ;th output of the middle layer of the critic, x i : i th state of the environment at time t, c\x and a-: : center and dispersion in the i th input of j th basis function, respectively, / : the number of nodes in the middle layer of the critic, n : number of the states of the system (see Fig. 2). 200 Robust Control, Theory and Applications Input layer Output layer Fig. 2. Structure of the critic. 2.1.2 Learning of parameters of critic Learning of parameters of the critic is done by back propagation method which makes prediction error r(t) go to zero. Updating rule of parameters are as follows, dco- (6) Here r/ c is a small positive value of learning coefficient. 2.2 Structure and learning of actor 2.2.1 Structure of actor Figure 3 shows the structure of the actor. The actor plays the role of controller and outputs the control signal, action a(t) , to the environment. The actor basically also consists of radial basis function network. The jth basis function of the middle layer node of the actor is as follows, y°(0 = exp 7K) 2 (7) «'(0=£"ry/(0' 7=1 (8) u 1 (t) = u n l + exp(-w'(t)) l-exp(-w'(0) (9) u(t) = u 1 (t) + n{t) (10) Here y a - : ;th node's output of the middle layer of the actor, c| and a^ : center and dispersion in zth input of ;th node basis function of the actor, respectively, coj : connection weight from ;th node of the middle layer to the output, u(t) : control input, n(t) : additive noise. A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 201 Input layer Middle layer Output layer i^(/) *<o Fig. 3. Structure of the actor. 2.2.2 Noise generator Noise generator let the output of the actor have the diversity by making use of the noise. It comes to realize the learning of the trial and error according to the results of performance of the system by executing the decided action. Generation of the noise n(t) is as follows, n(t) = n t = noise t • min(l,exp(-P(f )) , (ii) where noise t is uniformly random number of [-1 , l] , min ( •): minimum of • . As the P(t) will be bigger (this means that the action goes close to the optimal action), the noise will be smaller. This leads to the stable learning of the actor. 2.2.3 Learning of parameters of actor Parameters of the actor, co a - (j = 1, ••-,]) , are adjusted by using the results of executing the output of the actor, i.e. the prediction error r t and noise. A(D? du^t) ' dm] (12) 7] a (>0)is the learning coefficient. Equation (12) means that (-n t -r t ) is considered as an error, co a - is adjusted as opposite to sign of (-n t • f t ) . In other words, as a result of executing u(t) , e.g. if the sign of the additive noise is positive and the sign of the prediction error is positive, it means that positive additive noise is sucess, so the value of co] should be increased (see Eqs. (8)-(10)), and vice versa. 3. Controlled system, variable filter and sliding mode control 3.1 Controlled system This paper deals with next nth order nonlinear differential equation. » = f(x) + b(x)u, (13) 202 Robust Control, Theory and Applications y = x, (14) where x = [x,x,-",x^ n ~ 1 '] T is state vector of the system. In this paper, it is assumed that a part of states, y(= x) , is observable, u is control input, /(x), b(x) are unknown continuous functions. Object of the control system: To decide control input u which leads the states of the system to their targets x. We define the error vector e as follows, e = [e,e,---,e .(n-lUT (n-l)_„ (n-l)iT - [x-x d ,x-x d ,--- ,x The estimate vector of e, e , is available through the state variable filter (see Fig. 4). (15) 3.2 State variable filter Usually it is that not all the state of the system are available for measurement in the real system. In this work we only get the state x, that is, e, so we estimate the values of error vector e, i.e. e , through the state variable filter, Eq. (16) (Hang (1976) (see Fig. 4). <*>n'V V +<»n-lV -e, (z=0,..-,n-l) (16) Fig. 4. Internal structure of the state variable filter. 3.3 Sliding mode control Sliding mode control is described as follows. First it restricts states of the system to a sliding surface set up in the state space. Then it generates a sliding mode s (see in Eq. (18)) on the sliding surface, and then stabilizes the state of the system to a specified point in the state space. The feature of sliding mode control is good robustness. Sliding time-varying surface H and sliding scalar variable s are defined as follows, H:{e|s(e) = 0} (17) A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 203 s(e) T a e , where a n _ x -\ a = [a ,a 1 ,---,a n _ 1 ] T , and a n _ x ]) n 1 +oc n _ 1 ip n 2 Hurwitz, p is Laplace transformation variable. (18) -a is strictly stable in 4. Actor-critic reinforcement learning system using sliding mode control with state variable filter In this section, reinforcement learning system using sliding mode control with the state variable filter is explained. Target of this method is enhancing robustness which can not be obtained by conventional reinforcement. The method is almost same as the conventional actor-critic system except using the sliding variable s as the input to it inspite of the system states. In this section, we mainly explain the definition of the reward and the noise generation method. TV > Critic* < w Prediction reward Noise n Prediction error f a\t Actor ^ z-l control input ^6 Controlled system output x + y - Reward T sliding scalar variable s > reference *rf Reference error State variablefilter Fig. 5. Proposed reinforcement learning control system using sliding mode control with state variable filter. 4.1 Reward We define the reward r(t) to realize the sliding mode control as follows, r(t) = exp{-s(f) 2 (19) here, from Eq. (18) if the actor-critic system learns so that the sliding variable s becomes smaller, i.e., error vector e would be close to zero, the reward r(t) would be bigger. 4.2 Noise Noise n(t) is used to maintain diversity of search of the optimal input and to find the optimal input. The absolute value of sliding variable s is bigger, n(t) is bigger, and that of s is smaller, it is smaller. 204 Robust Control, Theory and Applications n(t) = z-n -exp -p (20) where, z is uniform random number of range [-1, 1]. n is upper limit of the perturbation signal for searching the optimal input u. p is predefined positive constant for adjusting. 5. Computer simulation 5.1 Controlled object To verify effectiveness of the proposed method, we carried out the control simulation using an inverted pendulum with dynamics described by Eq. (21) (see Fig. 6). mg6 = mglsirv0 - ju v 6 + T . Parameters in Eq. (21) are described in Table 1. (21) Fig. 6. An inverted pendulum used in the computer simulation. e joint angle - m mass 1.0 [kg] I length of the pendulum 1.0 [m] 8 gravity 9.8 [m/sec2] Mv coefficient of friction 0.02 T input torque - X = [0,0] observation vector - Table 1. Parameters of the system used in the computer simulation. 5.2 Simulation procedure Simulation algorithm is as follows, Step 1. Initial control input T is given to the system through Eq. (21). Step 2. Observe the state of the system. If the end condition is satisfied, then one trial ends, otherwise, go to Step 3. Step 3. Calculate the error vector e, Eq. (15). If only y(=x) , i.e., e is available, calculate e , the estimate value of through the state variable filters, Eq. (16). A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 205 Step 4. Calculate the sliding variable s, Eq. (18). Step 5. Calculate the reward r by Eq. (19). Step 6. Calculate the prediction reward P(t) and the control input u(t) , i.e., torque T by Eqs. (4) and (10), respectively. Step 7. Renew the parameters a>- ,a>j of the actor and the critic by Eqs. (6) and (12). Step 8. Set T in Eq. (21) of the system. Go to Step 2. 5.3 Simulation conditions One trial means that control starts at (0 Q/ ) = (7r/18[rad], [rad / sec] ) and continues the system control for 20[sec], and sampling time is 0.02[sec]. The trial ends if \0\ > n / 4 or controlling time is over 20[sec]. We set upper limit for output u x of the actor. Trial success means that 6 is in range [-tt/3 60,^/3 60] for last 10[sec]. The number of nodes of the hidden layer of the critic and the actor are set to 15 by trial and error (see Figs. (2)-( 3)). The parameters used in this simulation are shown in Table 2. a : sliding variable parameter in Eq. (18) 5.0 r] c : learning coefficient of the actor in Eqs. (6)-(A6) 0.1 rj a : learning coefficient of the critic in Eqs. (12) -A (7) 0.1 U max : Maximun value of the Torque in Eqs. (9)-(A3) 20 y : forgetting rate in Eq. (3) 0.9 Table 2. Parameters used in the simulation for the proposed system. 5.4 Simulation results Using subsection 5.2, simulation procedure, subsection 5.3, simulation conditions, and the proposed method mentioned before, the control simulation of the inverted pendulum Eq. (21) are carried out. 5.4.1 Results of the proposed method a. The case of complete observation The results of the proposed method in the case of complete observation, that is, 0, are available, are shown in Fig. 7. 0.4 "0.2 2 £-0.2 -0.4 HflUfi Position — Verocity — - V I „10- i- ° ■--io! Control signal — J \ 1 If 5 10 15 20 5 10 TIME [sec] TIME [sec] (a) 0,0 (b) Torque T q Fig. 7. Result of the proposed method in the case of complete observation (0,0 15 206 Robust Control, Theory and Applications b. The case of incomplete observation using the state variable filters . In the case that only 6 is available, we have to estimate 6 as 6 . Here, we realize it by use of the state variable filter (see Eqs. (22)-(23), Fig. 8). By trial and error, the parameters, co , co x , co 2 , of it are set to co = 100 , co x - 10 ,co 2 = 50 . The results of the proposed method with state variable filter in the case of incomplete observation are shown in Fig. 9. Fig. 8. State variable filter in the case of incomplete observation ( e ). 'o- — p + CO x p + COq ; CO lP '1 2 p + co x p + CO Q (22) (23) 0.4 ~0.2 CO -0.4 r Angular Rosition^ Angu ar Hosition Angular Velocity 20 JO _ ■1011 5 10 TIME [sec] 15 20 5 10 TIME [sec] Control signal —^ 15 (a) 0,6 (b) Torque T q Fig. 9. Results of the proposed method with the state variable filter in the case of incomplete observation (only 6 is available). c. The case of incomplete observation using the difference method Instead of the state variable filter in 5.4.1 B, to estimate the velocity angle, we adopt the commonly used difference method, like that, We construct the sliding variable s in Eq. (18) by using 0, the proposed method are shown in Fig. 10. (24) . The results of the simulation of A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 207 igular Position — par velocity — 5 10 ME [sec] 15 JO _ ■10 1 -20 ; i Control signal — I 5 10 TIME [sec] 15 20 (a) 6,6 (b) Torque T Fig. 10. Result of the proposed method using the difference method in the case of incomplete observation (only 6 is available). 5.4.2 Results of the conventional method. d. Sliding mode control method The control input is given as follows, U max ' u(t) = i — 1 1 max ' a = c6 + 6 ^max==20.0[N] if 6 -a > if • a < (25) Result of the control is shown in Fig. 11. In this case, angular, velocity angular, and Torque are all oscillatory because of the bang-bang control. TIME [seel (a) 0, 6 Timefsecl (b) Torque T Fig. 11. Result of the conventional (SMC) method in the case of complete observation (# ? q ). e. Conventional actor-critic method The structure of the actor of the conventional actor-critic control method is shown in Fig. 12. The detail of the conventional actor-critic method is explained in Appendix. Results of the simulation are shown in Fig. 13. 208 Robust Control, Theory and Applications Input layer Mi dale layer Output layer «!</) «(0 Fig. 12. Structure of the actor of the conventional actor-critic control method. 0.4 0.2, Angular Position - Angular Velocity 2 -0.2 ■0.4 Control signal =3 " r- T" W: o 5 10 TIME [sec] 15 20 5 10 TIME [sec] 15 20 (a) 6, (b) Torque T q Fig. 13. Result of the conventional (actor-critic) method in the case of complete observation (0,0). 0.4- ^ 0.2, CD N co 2 0; 1-0-2 -0.4 Angular Position — AnguarHositiq Angular Velocn Control signal — 5 10 TIME [sec] 15 20 5 10 TIME [sec] 15 (a) 0, (b) Torque T q Fig. 14. Result of the conventional PID control method in the case of complete observation {0,6). A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 209 / Conventional PID control method The control signal u(t) in the PID control is M (0 = -K p e(t) - Kj j%(f) • dt -K d -e(t) , (26) here, K = 45, Kj = 1, K d - = 10 . Fig. 14 shows the results of the PID control. 5.4.3 Discussion Table 3 shows the control performance, i.e. average error of 6,6, through the controlling time when final learning for all the methods the simulations have been done. Comparing the proposed method with the conventional actor-critic method, the proposed method is better than the conventional one. This means that the performance of the conventional actor- critic method hass been improved by making use of the concept of sliding mode control. Kinds of Average error Proposed method Conventional method Actor-Critic + SMC SMC PID Actor- Critic Complete observation Incomplete Observation ( 6 : available) Complete observation S.v.f. Difference fedt/t 0.3002 0.6021 0.1893 0.2074 0.4350 0.8474 fOdt/t 0.4774 0.4734 0.4835 1.4768 0.4350 1.2396 Table 3. Control performance when final learning (S.v.f. : state variable filter, Difference: Difference method). 0.2 0.15 0.1 © 0.05 -0.05 -0.1 l^^» -^-»l,-v+~ ,~+,-«+~ ^U~,~ - ,-.+ - ,-^ ^' w PID X \ i ^^5==-=^ -~- -^- \^_ 4 6 Time[sec] 10 Fig. 15. Comparison of the porposed method with incomplete observation, the conventional actor-critic method and PID method for the angle, 6 . 210 Robust Control, Theory and Applications Figure 15 shows the comparison of the porposed method with incomplete observation, the conventional actor-critic method and PID method for the angle, 6 . In this figure, the proposed method and PID method converge to zero smoothly, however the conventional actor-critic method does not converge. The comparison of the proposed method with PID control, the latter method converges quickly. These results are corresponding to Fig.16, i.e. the torque of the PID method converges first, the next one is the proposed method, and the conventional one does not converge. ii'llil'fiililiiiil'^il A m Incomplete state observation using State-filter RL+SMC actor-critic RL PID wwm W II |ll||||i!|| | i i.'\||')||M Li". | ■!! i'"l' 'll'ill'il '1,11 ,'! I, 10 Fig. 16. Comparison of the porposed method with incomplete observation, the conventional actor-critic method and PID method for the Torque, T . Incomplete state observation using State-filter RL+SMC Complete state observation RL+SMC Incomplete state observation using Differencial RL+SMC 0.2 0.15 0.1 0.05 -0.05 -0.1 0.5 1 1.5 2 2.5 TIME [sec] Fig. 17. The comparison of the porposed method among the case of the complete observation, the case with the state variable filter, and with the difference method for the angle, 6 . A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 211 Fig. 17 shows the comparison of the porposed method among the case of the complete observation, the case with the state variable filter, and with the difference method for the angle, . Among them, the incomplete state observation with the difference method is best of three, especially, better than the complete observation. This reason can be explained by Fig. 18. That is, the value of s of the case of the difference method is bigger than that of the observation of the velocity angle, this causes that the input gain becomes bigger and the convergence speed has been accelerated. Sliding using Velocity Sliding using Differencial 10 15 20 TIME [sec] Fig. 18. The values of the sliding variable s for using the velocity and the difference between the angle and 1 sampling past angle. 5.4.4 Verification of the robust performance of each method At first, as above mentioned, each controller was designed at m = 1.0 [kg] in Eq. (21). Next we examined the range of m in which the inverted pendulum control is success. Success is defined as the case that if \0\ <tt/45 through the last l[sec]. Results of the robust performance for change of m are shown in Table 4. As to upper/ lower limit of m for success, the proposed method is better than the conventional actor-critic method not only for gradually changing m smaller from 1.0 to 0.001, but also for changing m bigger from 1.0 to 2.377. However, the best one is the conventional SMC method, next one is the PID control method. 6. Conclusion A robust reinforcement learning method using the concept of the sliding mode control was mainly explained. Through the inverted pendulum control simulation, it was verified that the robust reinforcement learning method using the concept of the sliding mode control has good performance and robustness comparing with the conventional actor-critic method, because of the making use of the ability of the SMC method. 212 Robust Control, Theory and Applications The way to improve the control performance and to clarify the stability of the proposed method theoretically has been remained. Proposed method Conventional method Actor-Critic + SMC SMC PID Actor-Critic Complete observation Incomplete observ. + s.v.f.* Complete observation Complete observation Complete observation m-max [kg] 2.081 2.377 11.788 4.806 1.668 m-min [kg] 0.001 0.001 0.002 0.003 0.021 *(s.v.f.: state variable filter) Table 4. Robust control performance for change of m in Eq. (21). 7. Acknowledgement This work has been supported by Japan JSPS-KAKENHI (No.20500207 and No.20500277). 8. Appendix The structure of the critic of the conventional actor-critic control method is shown in Fig. 2. The number of nodes of the hidden layer of it is 15 as same as that of the proposed method. The prediction reward, P(t), is as follow, P(t) = 2>f ■ exp + (Al) The structure of actor is also similar with critic shown in Fig. 11. The output of the actor, u\t) , and the control input, u(t), are as follows, respectively, «) 2 (a \2 (A2) ' 6i> u l (t) = u n 1 + exp(-w (0) \-GX1p(-U,(t)) (A3) u(t) = u 1 (t) + n(t). (A4) The center, c c ei ,c c ^ p c a ei ,c a ^. of the critic and actor of the RBF network are set to equivalent distance in the range of -3 < c < 3 . The variance, <t^,<t^ .,0-^,0-^. of the critic and actor of A Robust Reinforcement Learning System Using Concept of Sliding Mode Control for Unknown Nonlinear Dynamical System 213 the RBF networks are set to be at equivalent distance in the range of [0 < a < 1] . The values mentioned above, particularly, near the original are set to close. The reward r(t) is set as Eq. (A5) in order it to maximize at {6,6) = (0,0) , r{t) = exp (e,f (e t f (A5) The learning of parameters of critic and actor are carried out through the back-propagation algorithm as Eqs. (A6)-(A7) . (r/ c ,r/ a > 0) A<=-7 C ~, (/ = 1,-,-/), (A6) oco i A^=rj a -n r r t -^-, (j = l,-,J). (A7) i 9. References K. Doya. (2000). " Reinforcement learning in continuous time and space", Neural Computation, 12(1), pp.219-245 C.C. Hang. (1976). " On state variable filters for adaptive system design", IEEEE Trans. Automatic Control Vo.21, No.6,874-876 C.C. Kung & T.H. Chen. (2005). "Observer-based indirect adaptive fuzzy sliding mode control with state variable filters for unknown nonlinear dynamical systems", Fuzzy Sets and Systems, Vol.155, pp.292-308 D.G. Luenberger. (1984). "Linear and Nonlinear Programming", Addison-Wesley Publishing Company, MA J. Morimoto & K. Doya. (2005) "Robust Reinforcement Learning", Neural Computation 17,335-359 M. Obayashi & T. Kuremoto & K. Kobayashi. (2008). "A Self -Organized Fuzzy-Neuro Reinforcement Learning System for Continuous State Space for Autonomous Robots", Proc. of International Conference on Computational Intelligence for Modeling, Control and Automation (CIMCA 2008), 552-559 M. Obayashi & N. Nakahara & T. Kuremoto & K. Kobayashi. (2009a). "A Robust Reinforcement Learning Using Concept of Slide Mode Control", The Journal of the Artificial Life and Robotics, Vol. 13, No. 2, pp.526-530 M. Obayashi & K. Yamada, T. & Kuremoto & K. Kobayashi. (2009b). "A Robust Reinforcement Learning Using Sliding Mode Control with State Variable Filters", Proceedings of International Automatic Control Conference (CACS 2009),CDROM J.J.E. Slotine & W. Li. (1991). "Applied Nonlinear Control ", Prentice-Hall, Englewood Cliffs, NJ R.S. Sutton & A.G. Barto. (1998). " Reinforcement Learning: An Introduction", The MIT Press. 21 4 Robust Control, Theory and Applications W.Y. Wang & M.L. Chan & C.C. James & T.T. Lee. (2002). " H ^ Tracking-Based Sliding Mode Control for Uncertain Nonlinear Systems via an Adaptive Fuzzy-Neural Approach", IEEE Trans, on Systems, Man, and Cybernetics, Vol.32, No. 4, August, pp.483-492 X. S. Wang & Y. H. Cheng & J. Q. Yi. (2007). "A fuzzy Actor-Critic reinforcement learning network", Information Sciences, 177, pp.3764-3781 C. Watkins & P. Dayan. (1992)."Q-learning," Machine learning, Vol.8, pp.279-292 K. Zhau & J.C.Doyle & K.Glover. (1996). "Robust optimal control", Englewood Cliffs NJ, Prentice Hall Part 4 Selected Trends in Robust Control Theory 10 Robust Controller Design: New Approaches in the Time and the Frequency Domains Vojtech Vesely, Danica Rosinova and Alena Kozakova Slovak University of Technology Slovak Republic 1. Introduction Robust stability and robust control belong to fundamental problems in control theory and practice; various approaches have been proposed to cope with uncertainties that always appear in real plants as a result of identification /modelling errors, e.g. due to linearization and approximation, etc. A control system is robust if it is insensitive to differences between the actual plant and its model used to design the controller. To deal with an uncertain plant a suitable uncertainty model is to be selected and instead of a single model, behaviour of a whole class of models is to be considered. Robust control theory provides analysis and design approaches based upon an incomplete description of the controlled process applicable in the areas of non-linear and time-varying processes, including multi input - multi output (MIMO) dynamic systems. MIMO systems usually arise as interconnection of a finite number of subsystems, and in general, multivariable centralized controllers are used to control them. However, practical reasons often make restrictions on controller structure necessary or reasonable. In an extreme case, the controller is split into several local feedbacks and becomes a decentralized controller. Compared to centralized full-controller systems such a control structure brings about certain performance deterioration; however, this drawback is weighted against important benefits, e.g. hardware, operation and design simplicity, and reliability improvement. Robust approach is one of useful ways to address the decentralized control problem (Boyd et al, 1994; Henrion et al, 2002; de Oliveira et al, 1999; Gyurkovics & Takacs, 2000; Ming Ge et al, 2002; Skogestad & Postlethwaite, 2005; Kozakova and Vesely, 2008; Kozakova et al, 2009a) . In this chapter two robust controller design approaches are presented: in the time domain the approach based on Linear (Bilinear) matrix inequality (LMI, BMI), and in the frequency domain the recently developed Equivalent Subsystem Method (ESM) (Kozakova et al., 2009b). As proportional-integral-derivative (PID) controllers are the most widely used in industrial control systems, this chapter focuses on the time- and frequency domain PID controller design techniques resulting from both approaches. The development of Linear Matrix Inequality (LMI) computational techniques has provided an efficient tool to solve a large set of convex problems in polynomial time (e.g. Boyd et al., 1994). Significant effort has been therefore made to formulate crucial control problems in 21 8 Robust Control, Theory and Applications algebraic way (e.g. Skelton et al., 1998), so that the numerical LMI solution can be employed. This approach is advantageously used in solving control problems for linear systems with convex (affine or polytopic) uncertainty domain. However, many important problems in linear control design, such as decentralized control, simultaneous static output feedback (SOF) or more generally - structured linear control problems have been proven as NP hard (Blondel & Tsitsiklis, 1997). Though there exist solvers for bilinear matrix inequalities (BMI), suitable to solve e.g. SOF, they are numerically demanding and restricted to problems of small dimensions. Intensive research has been devoted to overcome nonconvexity and transform the nonconvex or NP-hard problem into convex optimisation problem in LMI framework. Various techniques have been developed using inner or outer convex approximations of the respective nonconvex domains. The common tool in both inner and outer approximation is the use of linearization or convexification. In (Han & Skelton, 2003; de Oliveira et al., 1999), the general convexifying algorithm for the nonconvex function together with potential convexifying functions for both continuous and discrete-time case have been proposed. Linearization approach for continuous and discrete-time system design was independently used in (Rosinova & Vesely, 2003; Vesely, 2003). When designing a (PID) controller, the derivative part of the controller causes difficulties when uncertainties are considered. In multivariable PID control schemes using LMI developed recently (Zheng et al., 2002), the incorporation of the derivative part requires inversion of the respective matrix, which does not allow including uncertainties. Another way to cope with the derivative part is to assume the special case when output and its derivative are state variables, robust PID controller for first and second order SISO systems are proposed for this case in (Ming Ge et al., 2002). In Section 2, the state space approach to the design of (decentralized or multi-loop) PID robust controllers is proposed for linear uncertain system with guaranteed cost using a new quadratic cost function. The major contribution is in considering the derivative part in robust control framework. The resulting matrix inequality can be solved either using BMI solver, or using linearization approach and following LMI solution. The frequency domain design techniques have probably been the most popular among the practitioners due to their insightfulness and link to the classical control theory. In combination with the robust approach they provide a powerful engineering tool for control system analysis and synthesis. An important field of their implementation is control of MIMO systems, in particular the decentralized control (DC) due to simplicity of hardware and information processing algorithms. The DC design proceeds in two main steps: 1) selection of a suitable control configuration (pairing inputs with outputs); 2) design of local controllers for individual subsystems. There are two main approaches applicable in Step 2: sequential (dependent) design, and independent design. When using sequential design local controllers are designed sequentially as a series controller, hence information about " lower level" controllers is directly used as more loops are closed. Main drawbacks are lack of failure tolerance when lower level controllers fail, strong dependence of performance on the loop closing order, and a trial-and-error design process. According to the independent design, local controllers are designed to provide stability of each individual loop without considering interactions with other subsystems. The effect of interactions is assessed and transformed into bounds for individual designs to guarantee stability and a desired performance of the full system. Main advantages are direct design of local controllers with no need for trial and error; the limitation consists in that information Robust Controller Design: New Approaches in the Time and the Frequency Domains 21 9 about controllers in other loops is not exploited, therefore obtained stability and performance conditions are only sufficient and thus potentially conservative. Section 3 presents a frequency domain robust decentralized controller design technique applicable for uncertain systems described by a set of transfer function matrices. The core of the technique is the Equivalent Subsystems Method - a Nyquist-based DC design method guaranteeing performance of the full system (Kozakova et al., 2009a; 2009b). To guarantee specified performance (including stability), the effect of interactions is assessed using a selected characteristic locus of the matrix of interactions further used to reshape frequency responses of decoupled subsystems thus generating so-called equivalent subsystems. Local controllers of equivalent subsystems independently tuned to guarantee specified performance measure value in each of them constitute the decentralized (diagonal) controller; when applied to real subsystems, the resulting controller guarantees the same performance measure value for the full system. To guarantee robust stability over the specified operating range of the plant, the M-A stability conditions are used (Skogestad & Postlethwaite, 2005; Kozakova et al., 2009a, 2009b). Two versions of the robust DC design methodology have been developed: a the two-stage version (Kozakova & Vesely, 2009; Kozakova et al. 2009a), where robust stability is achieved by additional redesign of the DC parameters; in the direct version, robust stability conditions are integrated in the design of local controllers for equivalent subsystems. Unlike standard robust approaches, the proposed technique allows considering full nominal model thus reducing conservatism of robust stability conditions. Further conservatism relaxing is achieved if the additive affine type uncertainty description and the related M a f- Q stability conditions are used (Kozakova & Vesely, 2007; 2008). In the sequel, X > denotes positive definite matrix; * in matrices denotes the respective transposed term to make the matrix symmetric; I denotes identity matrix and denotes zero matrix of the respective dimensions. 2. Robust PID controller design in the time domain In this section the PID control problem formulation via LMI is presented that is appropriate for poly topic uncertain systems. Robust PID control scheme is then proposed for structured control gain matrix, thus enabling decentralized PID control design. 2.1 Problem formulation and preliminaries Consider the class of linear affine uncertain time-invariant systems described as: Sx(t) = (A + SA)x(t) + (B + SB)u(t) y(t) = Cx(t) (1) where Sx(t) = x(t) for continuous-time system Sx(t ) = x(t + 1) for discrete-time system x(t)eR n ,u(t)eR m ,y(t)eR are state, control and output vectors respectively; A, B, C are known constant matrices of the respective dimensions corresponding to the nominal system, SA,SB are matrices of uncertainties of the respective dimensions. The affine uncertainties are assumed 220 Robust Control, Theory and Applications SA{t) = f jYj A j ,SB{t) = f j y j B j (2) 7=1 ;=i where /.</•</• are unknown uncertainty parameters; A ,B ,; = 1,2,...,/? are constant matrices of uncertainties of the respective dimensions and structure. The uncertainty domain for a system described in (1), (2) can be equivalently described by a polytopic model given by its vertices {(A^CUA^C) (A N ,B N ,C)} , N = 2? (3) The (decentralized) feedback control law is considered in the form u(t) = FCx(t) (4) where F is an output feedback gain matrix. The uncertain closed-loop polytopic system is then Sx(t) = A c (a)x(t) (5) where ( N N 1 A c (a)e X^ A Cf Z a i =1 ' a i^°\> ( 6 ) A Cl =A l+ B l FC. To assess the performance, a quadratic cost function known from LQ theory is frequently used. In practice, the response rate or overshoot are often limited, therefore we include the additional derivative term for state variable into the cost function to damp the oscillations and limit the response rate. oo J c = j[x(t) T Qx(t) + u(t) T Ru(t) + Sx(tf SSx(t)]dt for a continuous-time and (7) Jd = Z W0 T Q x (0 + u(t) T Ru(t) + Sx(t) T SSx(t)] for a discrete-time system (8) fc=0 where Q,S eR nxn ,ReR mxm are symmetric positive definite matrices. The concept of guaranteed cost control is used in a standard way: let there exist a feedback gain matrix Fo and a constant Jo such that 7^/o (9) holds for the closed loop system (5), (6). Then the respective control (4) is called the guaranteed cost control and the value of Jo is the guaranteed cost. The main aim of Section 2 of this chapter is to solve the next problem. Problem 2.1 Find a (decentralized) robust PID control design algorithm that stabilizes the uncertain system (1) with guaranteed cost with respect to the cost function (7) or (8). Robust Controller Design: New Approaches in the Time and the Frequency Domains 221 We start with basic notions concerning Lyapunov stability and convexifying functions. In the following we use D-stability concept (Henrion et aL, 2002) to receive the respective stability conditions in more general form. Definition 2.1 (D-stability) Consider the D-domain in the complex plain defined as D = {s is complex number : "1" s hi hi hi r n_ T s <0} The considered linear system (1) is D-stable if and only if all its poles lie in the D-domain. (For simplicity, we use in Def. 2.1 scalar values of parameters rij, in general the stability domain can be defined using matrix values of parameters r^ with the respective dimensions.) The standard choice of r^ is rn = 0, ri2 = 1, r22 = for a continuous-time system; rn = -1, ri2 = 0, r22 = 1 for a discrete-time system, corresponding to open left half plane and unit circle respectively. The quadratic D-stability of uncertain system is equivalent to the existence of one Lyapunov function for the whole uncertainty set. Definition 2.2 (Quadratic D-stability) The uncertain system (5) is quadratically D-stable if and only if there exists a symmetric positive definite matrix P such that r 12 PA c (a) + h 2 A[(a)P + r n P + r 22 A T c (a)PA c (a) < (10) To obtain less conservative results than using quadratic stability, a robust stability notion is considered based on the parameter dependent Lyapunov function (PDLF) defined as P(a) = ^afi where P { = Pj > (11) Definition 2.3 (deOliveira et aL, 1999) System (5) is robustly D-stable in the convex uncertainty domain (6) with parameter- dependent Lyapunov function (11) if and only if there exists a matrix P(a) = P(a) T > such that r 12 P(a)A c (a) + r* 12 A T c {a)P{a) + r n P(a)+ r 22 A£(tf)P(a)A c (a) < (12) for all a such that A c (a) is given by (6). Now recall the sufficient robust D-stability condition proposed in (Peaucelle et aL, 2000), proven as not too conservative (Grman et aL, 2005). Lemma 2.1 If there exist matrices E e R nxn , G e R nxn and N symmetric positive definite matrices P { e R nxn such that for all i = 1, . . ., N: r„P+Ar-E T +EA t a h^-E + A^G rl 2 P l -E T + G T A Cl r 22 ^.-(G + G T ) <0 (13) then uncertain system (5) is robustly D-stable. 222 Robust Control, Theory and Applications Note that matrices E and G are not restricted to any special form; they were included to relax the conservatism of the sufficient condition. To transform nonconvex problem of structured control (e.g. output feedback, or decentralized control) into convex form, the convexifying (linearizing) function can be used (Han&Skelton, 2003; deOliveira et al., 2000; Rosinova&Vesely, 2003; Vesely, 2003). The respective potential convexifying function for X -1 and XPVX has been proposed in the linearizing form: the linearization of X -1 e R nxn about the value X k > is 0(X~\X k ) = XI 1 -Xl\X-X k )Xt (14) the linearization of XWX e R nxn about X k is ^(XWX, X k ) = -X k WX k + XWX k + X k WX (15) Both functions defined in (14) and (15) meet one of the basic requirements on convexifying function: to be equal to the original nonconvex term if and only if Xk = X. However, the question how to choose the appropriate nice convexifying function remains still open. 2.2 Robust optimal controller design In this section the new design algorithm for optimal control with guaranteed cost is developed using parameter dependent Lyapunov function and convexifying approach employing iterative procedure. The proposed control design approach is based on sufficient stability condition from Lemma 2.1. The next theorem provides the new form of robust stability condition for linear uncertain system with guaranteed cost. Theorem 2.1 Consider uncertain linear system (1), (2) with static output feedback (4) and cost function (7) or (8). The following statements are equivalent: i. Closed loop uncertain system (5) is robustly D-stable with PDLF (11) and guaranteed cost with respect to cost function (7) or (8): / < J = x T (0)P(a)x(0) . ii. There exist matrices P(a) > defined by (11) such that r 12 P(a)A c (a) + ri 2 A T c {a)P{a) + r n P(a) + r 22 A T c (a)P(a)A c (a) + +Q + C T F T RFC + A T c (a)SA c (a) < iii. There exist matrices P(a) > defined by (11) and matrices H, G and F of the respective dimensions such that r n P(a) + A T ci (a)H T + HA ci (a) + Q + C T F T RFC ri 2 P(a) - H T + G T A ci (a) r 22 P(a) -(G + G T ) + S < (17) A ci = (A f + BiFC) denotes the i-th closed loop system vertex. Matrix F is the guaranteed cost control gain for the uncertain system (5), (6). Proof. For brevity the detail steps of the proof are omitted where standard tools are applied, (i) <^> (ii): the proof is analogous to that in (Rosinova, Vesely, Kucera, 2003). The (ii) =>(i) is shown by taking V(t) = x(t)P(a)x(t) as a candidate Lyapunov function for (5) and writing SV(t) , where Robust Controller Design: New Approaches in the Time and the Frequency Domains 223 SV(t) = V(t) for continuous-time system SV(t) = V(t + 1) - V(t) for discrete-time system SV (t) = r* 2 Sx(t) T P{a)x{t) + r 12 x(t) T P(a)Sx(t) + r n x(t) T P{a)x{t) + r 22 Sx(t) T P{a)Sx{t) (18) Substituting for Sx from (5) to (18) and comparing with (16) provides D -stability of the considered system when the latter inequality holds. The guaranteed cost can be proved by summing or integrating both sides of the following inequality for t from to go: SV(t) < -x(t) T [Q + C T F T RFC + A T c (a)SA c (a)]x(t) The (i) =>(ii) can be proved by contradiction. (ii) <^> (iii): The proof follows the same steps to the proof of Lemma 2.1: (iii) =>(ii) is proved in standard way multiplying both sides of (17) by the full rank matrix (equivalent transformation) : [i A T c (a)]{lh.s.(17)} I A c (a) <o. (ii) ^>(iii) follows from applying a Schur complement to (16) rewritten as r 12 P(a)A c (a) + r* 12 A T c (a)P(a) + Q + C T F T RFC + r n P(a) + A T C (a)[r 22 P(a) + S]A C (a) < Therefore X n X 12 x 12 x 22 ^ : where X n = r n P(a) + r 12 P(a)A c (a) + rl 2 A T c (a)P(a) + Q + C T F T RFC X 12 =A T c (a)[r 22 P(a) + S] X 22 =-[r 22 P(a) + S] which for H = r 12 P(a), G= [r 22 P(a) + S] gives (17). The proposed guaranteed cost control design is based on the robust stability condition (17). Since the matrix inequality (17) is not LMI when both P(a) and F are to be found, we use the inner approximation for the continuous time system applying linearization formula (15) together with using the respective quadratic forms to obtain LMI formulation, which is then solved by iterative procedure. 2.3 PID robust controller design for continuous-time systems Control algorithm for PID is considered as u(t) = K p y(t) + Kjjy(t)dt + F d C d x(t) (19) The proportional and integral term can be included into the state vector in the common way t defining the auxiliary state z = \y(t) , i.e. z(t) = y(t) = Cx(t) . Then the closed-loop system for o PI part of the controller is 224 Robust Control, Theory and Applications A + SA 0] X \B + SBl C z + U (t) and u(t) = FCx(t) + F d C d x(t) (20) where FCx(t) and F d C d x(t) correspond respectively to the PI and D term of PID controller. The resulting closed loop system with PID controller (19) is then x n (t) = A c (a)x n (t) + B(a)[F d C d 0]*„(t) (21) where the PI controller term is included in A c (a) . (For brevity we omit the argument t) To simplify the denotation, in the following we consider PD controller (which is equivalent to the assumption, that the I term of PID controller has been already included into the system dynamics in the above outlined way) and the closed loop is described by x(t) = A c (a)x(t) + B(a)F d C d x(t) Let us consider the following performance index 7. = J[* *f Q + C T F T RFC S dt (22) (23) which formally corresponds to (7). Then for Lyapunov function (11) we have the necessary and sufficient condition for robust stability with guaranteed cost in the form (16), which for continuous time system can be rewritten as: [x x? Q + C T F T RFC P(a) <0 (24) P(a) S The main result on robust PID control stabilization is summarized in the next theorem. Theorem 2.2 Consider a continuous uncertain linear system (1), (2) with PID controller (19) and cost function (23). The following statements are equivalent: i Closed loop system (21) is robustly D-stable with PDLF (11) and guaranteed cost with respect to cost function (23): J<J = x T (0)P(a)x(0) . ii There exist matrices P(a) > defined by (11), and H, G, F and Fd of the respective dimensions such that A T Cj H T + HA ci +Q + C T F T RFC P-MlH + G T A r ; -M T di G-G T M dl+ S_ <0 (25) A ci = (A t + BiFC) denotes the i-th closed loop system vertex, Mdi includes the derivative part of the PID controller: M di =1- B i F d C d . Proof. Owing to (22) for any matrices H and G: {-x T H - x T G T )(x-A c (a)x - B(a)F d C d x) + +(x - A c (a)x - B(a)F d C d x) T (h t x - G*) = (26) Robust Controller Design: New Approaches in the Time and the Frequency Domains 225 Summing up the l.h.s of (26) and (24) and taking into consideration linearity w.r.t. a we get condition (25). Theorem 2.2 provides the robust stability condition for the linear uncertain system with PID controller. Notice that the derivative term does not appear in the matrix inversion and allows including the uncertainty in control matrix B into the stability condition. Considering PID control design, there are unknown matrices H, G, F and Fd to be solved from (25). (Recall that A Ci = (A t + B t FC) , M dl ■= I - B t F d C d .) Then, inequality (25) is bilinear with respect to unknown matrices and can be solved either by BMI solver, or by linearization approach using (15) to cope with the respective unknown matrices products. For the latter case the PID iterative control design algorithm based on LMI (4x4 matrix) has been proposed. The resulting closed loop system with PD controller is m = (I- W,,)" 1 (A + B,FC)x(t) , i=l N (27) The extension of the proposed algorithm to decentralized control design is straightforward since the respective F and Fd matrices are assumed as being of the prescribed structure, therefore it is enough to prescribe the decentralized structure for both matrices. 2.4 PID robust controller design for discrete-time systems Control algorithm for discrete-time PID (often denoted as PSD controller) is considered as u(k) = k P e(k) + k^e(k) + k D [e(k) - e(k - 1)] z=0 (28) control error e(k) = w- y(k) ; discrete time being denoted for clarity as k instead of t. PSD description in state space: z(k + l) = 1 1 m- e(k) = A R z(k) + B R e(k) (29) u(k) = [k D kj - k D ]z(k) + (k P + k l+ k D )e(k) Combining (1) for t^k and (29) the augmented closed loop system is received as x(k + l) A + SA x(k) B + SB z(k + l) _ -b r c a r _ lm\ K 2 =(l : P + k T +l d) K 1= [k D k r -*d]- [-K 2 K,] C I x(kj z(k)_ (30) Note that there is a significant difference between PID (19) and PSD (28) control design problem: for continuous time PID structure results in closed loop system that is not strictly proper which complicates the controller design, while for discrete time PSD structure, the control design is formulated as static output feedback (SOF) problem therefore the respective techniques to SOF design can be applied. In this section an algorithm for PSD controller design is proposed. Theorem 2.1 provides the robust stability condition for the linear time varying uncertain system, where a constrained control structure can be assumed: considering A Ci = (A z + B Z FC) we have SOF problem formulation which is also the case of discrete time PSD control structure for 226 Robust Control, Theory and Applications F = [-(fc p +fcj+fc D ) k E -k D ] (see (30)); (taking block diagonal structure of feedback matrix gain F provides decentralized controller). Inequality (17) is LMI for stability analysis for unknown H, G and Pi, however considering control design, having one more unknown matrix F in A Ci = (A z + B Z FC) , the inequality (17) is no more LMI. Then, to cope with the respective unknown matrix products the inner approximation approach can be used, when the resulting LMI is sufficient for the original one to hold. The next robust output feedback design method is based on (17) using additional constraint on output feedback matrix and the state feedback control design approach proposed respectively in (Crusius and Trofino, 1999; deOliveira et al., 1999). For stabilizing PSD control design (without considering cost function) we have the following algorithm (taking H=0, Q=0, R=0, S=0). PSD controller design algorithm Solve the following LMI for unknown matrices F, M, G and Pi of appropriate dimensions, the Pi being symmetric, positive definite, M, G being any matrices with corresponding dimensions: g t a[ + c t k t bJ ^G + B^C -G-G T + P + S <0 (31) 3>0, i = l,...,N MC = CG Compute the corresponding output feedback gain matrix (32) F = KM~ (33) where F = [-(fc K + k u +k Di ) k c ■k Di ] The algorithm above is quite simple and often provides reasonable results. 2.5 Examples In this subsection the major contribution of the proposed approach: design of robust controller with derivative feedback is illustrated on the examples. The results obtained using the proposed new iterative algorithm based on (25) to design the PD controller are provided and discussed. The impact of matrix S choice is studied as well. We consider affine models of uncertain system (1), (2) with symmetric uncertainty domain: -q,Sj=q Example 2.1 Consider the uncertain system (1), (2) where -4.365 -0.6723 -0.3363 7.0880 -6.5570 -4.6010 -2.4100 7.5840 -14.3100 2.3740 0.7485 1.3660 3.4440 0.9461 -9.6190 C = C, 1 uncertainty parameter q=l; uncertainty matrices Robust Controller Design: New Approaches in the Time and the Frequency Domains 227 -0.5608 0.8553 0.5892 2.3740 0.7485 A = 0.6698 -1.3750 -0.9909 Bi = 1.3660 3.4440 3.1917 1.7971 -2.5887 0.9461 -9.6190 " 0.6698 -1.3750 -0.9909" " 0.1562 0.1306 A 2 = -2.8963 -1.5292 10.5160 B 2 = -0.4958 4.0379 -3.5777 2.8389 1.9087 -0.0306 0.8947 The uncertain system can be described by 4 vertices; corresponding maximal eigenvalues in the vertices of open loop system are respectively: -4.0896 ± 2.1956i; -3.9243; 1.5014; -4.9595. Notice, that the open loop uncertain system is unstable (positive eigenvalue in the third vertex). The stabilizing optimal PD controller has been designed by solving matrix inequality (25). Optimality is considered in the sense of guaranteed cost w.r.t. cost function (23) with matrices R = I 2x2 , Q = 0.001 * I 3x3 . The results summarized in Tab.2.1 indicate the differences between results obtained for different choice of cost matrix S respective to a derivative of x. s Controller matrices F (proportional part) Fd (derivative part) Max eigenvalues in vertices le-6 *I F = "-1.0567 -0.5643" -2.1825 -1.4969 "-0.3126 -0.2243" -0.0967 0.0330 -4.8644 -2.4074 -3.8368 ± 1.1165 i -4.7436 0.1*1 F = "-1.0724 -0.5818" -2.1941 -1.4642 "-0.3227 -0.2186" -0.0969 0.0340 -4.9546 -2.2211 -3.7823 ± 1.4723 i -4.7751 Table 2.1 PD controllers from Example 2.1. Example 2.2 Consider the uncertain system (1), (2) where A = -2.9800 0.9300 -0.9900 -0.2100 0.0350 0.3900 -5.5550 -0.0340" "-0.0320" -0.0011 1 B = C = -1.8900 -1.6000 10" 1 1.5 0" "0" B = 228 Robust Control, Theory and Applications The results are summarized in Tab.2.2 for R = 1, Q = 0.0005 * I 4x4 for various values of cost function matrix S. As indicated in Tab.2.2, increasing values of S slow down the response as assumed (max. eigenvalue of closed loop system is shifted to zero). s Ojmax Max. eigenvalue of closed loop system le-8 *I 1.1 -0.1890 0.1*1 1.1 -0.1101 0.2*1 1.1 -0.0863 0.29 *I 1.02 -0.0590 Table 2.2 Comparison of closed loop eigenvalues (Example 2.2) for various S. 3. Robust PID controller design in the frequency domain In this section an original frequency domain robust control design methodology is presented applicable for uncertain systems described by a set of transfer function matrices. A two- stage as well as a direct design procedures were developed, both being based on the Equivalent Subsystems Method - a Nyquist-based decentralized controller design method for stability and guaranteed performance (Kozakova et al., 2009a;2009b), and stability conditions for the M-A structure (Skogestad & Postlethwaite, 2005; Kozakova et al., 2009a, 2009b). Using the additive affine type uncertainty and related M a f-Q structure stability conditions, it is possible to relax conservatism of the M-A stability conditions (Kozakova & Vesely, 2007). 3.1 Preliminaries and problem formulation Consider a MIMO system described by a transfer function matrix G(s) e R mxm f and a controller R(s) e R mxm m the standard feedback configuration (Fig. 1); w, u, y, e, d are respectively vectors of reference, control, output, control error and disturbance of compatible dimensions. Necessary and sufficient conditions for internal stability of the closed-loop in Fig. 1 are given by the Generalized Nyquist Stability Theorem applied to the closed-loop characteristic polynomial detF(s) = det[I + Q(s)] where Q(s) = G(s)R(s) e R mxm [ s the open-loop transfer function matrix. (34) w . R(s) *0^+ G(s) JU Fig. 1. Standard feedback configuration The following standard notation is used: D - the standard Nyquist D-contour in the complex plane; Nyquist plot of g(s) - the image of the Nyquist contour under g(s);N[k,g(s)] - the number of anticlockwise encirclements of the point (k, jO) by the Nyquist plot of g(s). Characteristic functions of Q(s) are the set of m algebraic functions q i (s),i = l,...,m given as Robust Controller Design: New Approaches in the Time and the Frequency Domains 229 de%(s)I m - Q(s)] = i = l,...,m (35) Characteristic loci (CL) are the set of loci in the complex plane traced out by the characteristic functions of Q(s), \/seD . The closed-loop characteristic polynomial (34) expressed in terms of characteristic functions of Q(s) reads as follows m detF(s) = det[7 + Q(s)] = l\[l + cj,(s)] (36) Theorem 3.1 (Generalized Nyquist Stability Theorem) The closed-loop system in Fig. 1 is stable if and only if a. detF(s)*0 VsgD b. N[0,detF(s)] = 2>{0,[1 + qi (s)]} = n q i=l (37) where F(s) = (I + Q(s)) and n q is the number of unstable poles of Q(s). Let the uncertain plant be given as a set IJoiN transfer function matrices 77 = {G fc (s)}, k = 1,2,..., N where G k (s) = {gUs)} (38) The simplest uncertainty model is the unstructured uncertainty, i.e. a full complex perturbation matrix with the same dimensions as the plant. The set of unstructured perturbations Du is defined as follows D u := {E(jcd) : cr max [E(^)] < l(a>), t{a>) = max cr max [E(^)]} (39) k where £{co) is a scalar weight function on the norm-bounded perturbation A(s)<=R mxm , cr max [zl(7^)]<l over given frequency range, 0" max (-) is the maximum singular value of (.), i.e. E(jco) = £(cd)A(jcd) (40) For unstructured uncertainty, the set 77 can be generated by either additive (E a ), multiplicative input (E,-) or output (E ) uncertainties, or their inverse counterparts (E za , En, Efo), the latter used for uncertainty associated with plant poles located in the closed right half-plane (Skogestad & Postlethwaite, 2005). Denote G(s) any member of a set of possible plants I7 k ,k = a,i,o,ia,ii,io; G (s) the nominal model used to design the controller, and ^{co) the scalar weight on a normalized perturbation. Individual uncertainty forms generate the following related sets IJ k : Additive uncertainty: n a := (G(s) : G(s) = G (s) + E a (s), E a {jm) < t a (a,)^ja,)\ / fl H = max<T max [G S: (H-G (H], k = l,2,...,N k (41) 230 Robust Control, Theory and Applications Multiplicative input uncertainty: 77, := {G(s) : G(s) = G (s)[l + E,(s)], £,(;«.) < <,(^K|ffl)) / > H = max<T max {G - 1 0«)[G t O ft >)-G (/ ft >)]}, * = 1,2,...,N Multiplicative output uncertainty: n := {G(s) : G(s) = [I + E o (s)]G (s), E Q (jm) < ( (jw)A(jw)} l (m) = max a m3X {[G k (j a )-G (j a )]G- \j a )}, k = l,2,...,N k Inverse additive uncertainty n ia := {G(s) : G(s) = G (s)[I -E, fl ( S )G (;«)]- 1 , £,,(/«) < l ia (co)A(ja>)} l m (w) = max ff^flGoOffljr 1 -^O)]" 1 ), * = 1,2 N k Inverse multiplicative input uncertainty 77„ := {G(s) : G(s) = G (s)[I - E tt (s)]-\ £„(;«) < e ti (co)A(jco)} £ u (w) = max <x max {I -[G k {jm)T\G (/«)]}, fc = l,2 N Inverse multiplicative output uncertainty: n w := {G(s) : G{s) = [I - E <o (s)]- 1 G (s), £,„(;«) < ^(a>)2l(;fl>)} f,>) = max ff^d-IGoOfflHIG^i^r 1 }, * = 1,2 N (42) (43) (44) (45) (46) Standard feedback configuration with uncertain plant modelled using any above unstructured uncertainty form can be recast into the M-A structure (for additive perturbation Fig. 2) where M(s) is the nominal model and A(s) e R mxm is the norm-bounded complex perturbation. If the nominal closed-loop system is stable then M(s) is stable and A(s) is a perturbation which can destabilize the system. The following theorem establishes conditions on M(s) so that it cannot be destabilized by A(s) (Skogestad & Postlethwaite, 2005). R(8) la yd A(s) -, Go(s) "D •6 ^> A(s) U A — ► M(s) Ya Fig. 2. Standard feedback configuration with unstructured additive uncertainty (left) recast into the M-A structure (right) Theorem 3.2 (Robust stability for unstructured perturbations) Assume that the nominal system M(s) is stable (nominal stability) and the perturbation A(s) is stable. Then the M-A system in Fig. 2 is stable for all perturbations A ( S ) ' °max ( A ) < 1 if and onl Y if Robust Controller Design: New Approaches in the Time and the Frequency Domains 231 <WM(;«)]<1, Vo) (47) For individual uncertainty forms M(s) = l k M k (s), k = a,i,o,ia,ii,io ; the corresponding matrices M k (s) are given below (disregarding the negative signs which do not affect resulting robustness condition); commonly, the nominal model G (s) is obtained as a model of mean parameter values. M(s) = £ a (s)R(s)[I + G^Ris)]' 1 = £ a (s)M a (s) additive uncertainty (48) M(s) = £ i (s)R(s)[I + G (s)R(s)]~ 1 G (s) = £ i (s)M i (s) multiplicative input uncertainty (49) M(s) = £ (s)G (s)R(s)[I + G (s)R(s)]~ 1 = £ (s)M (s) multiplicative output uncertainty (50) M(s) = £ ia (s)[I + Gq^R^s^G^s) = £ ia (s)M ia (s) inverse additive uncertainty (51) M(s) = I a (s) [I + R(s)G (s)] _1 = In (s)Mu (s) inverse multiplicative input uncertainty (52) M(s) = £ io (s)[I + G (s)R(s)]~ 1 = £ io (s)M io (s) inverse multiplicative output uncertainty (53) Conservatism of the robust stability conditions can be reduced by structuring the unstructured additive perturbation by introducing the additive affine-type uncertainty E a r(s) that brings about new way of nominal system computation and robust stability conditions modifiable for the decentralized controller design as (Kozakova & Vesely, 2007; 2008). v I jinxm ■ E af (s) = Z G i( s )li (54) where Gi(s) e R mxm t i=0,l, . . ., p are stable matrices, p is the number of uncertainties defining 2 p polytope vertices that correspond to individual perturbed models; c\i are polytope parameters. The set IJ a r generated by the additive affine-type uncertainty (E a f) is v n af '= l G ( S ) : G ( S ) = G o( S ) + E af E af = Z G *( S )^' °\i G< ^min^fmax >> limin + ^max = °l ( 55 ) i=\ where G (s) is the „afinne // nominal model. Put into vector-matrix form, individual perturbed plants (elements of the set IT a r ) can be expressed as follows G(s) = G (s) + [I ql ...I qp ] Gi(s) G P (s) -G (s) + QG u (s) (56) (mxp)xm where Q = [I qi ... ij e R'"^ , \=^\ m%m , G„(s) = [G 1 ...G p f eR Standard feedback configuration with uncertain plant modelled using the additive affine type uncertainty is shown in Fig. 3 (on the left); by analogy with previous cases, it can be recast into the M a f - Q structure in Fig. 3 (on the right) where 232 Robust Control, Theory and Applications M af = G U R(I + GoR)" 1 = G U (I + RG )" a R (57) R(s) Yq r* G u (s) -+ q -i U Q Go(s) — CH y ^> UQ Maf(s) Yq Fig. 3. Standard feedback configuration with unstructured af fine-type additive uncertainty (left), recast into the M a f -Q structure (right) Similarly as for the M-A system, stability condition of the M fl r - Q system is obtained as <(M af Q) <1 (58) Using singular value properties, the small gain theorem, and the assumptions that % = | <7imin| H^zmax| anc * tne nom i na l model M a f(s) is stable, (58) can further be modified to yield the robust stability condition K(M af )q Jp<l (59) The main aim of Section 3 of this chapter is to solve the next problem. Problem 3.1 Consider an uncertain system with m subsystems given as a set of N transfer function matrices obtained in N working points of plant operation, described by a nominal model G (s) and any of the unstructured perturbations (41) - (46) or (55). Let the nominal model G (s) can be split into the diagonal part representing mathematical models of decoupled subsystems, and the off-diagonal part representing interactions between subsystems G (s) = G d (s) + G m (s) where G i (s) = diag{G l (8)) mm ,detG i (s)*0 Vs G m (s) = G (s)-G d (s) A decentralized controller R(s) = diag{R,(s)} mxm . detR(s) * Vs e D (60) (61) (62) is to be designed with R^s) being transfer function of the i-th local controller. The designed controller has to guarantee stability over the whole operating range of the plant specified by either (41) - (46) or (55) (robust stability) and a specified performance of the nominal model (nominal performance). To solve the above problem, a frequency domain robust decentralized controller design technique has been developed (Kozakova & Vesely, 2009; Kozakova et. al., 2009b); the core of it is the Equivalent Subsystems Method (ESM). Robust Controller Design: New Approaches in the Time and the Frequency Domains 233 3.2 Decentralized controller design for performance: equivalent subsystems method The Equivalent Subsystems Method (ESM) an original Nyquist-based DC design method for stability and guaranteed performance of the full system. According to it, local controller designs are performed independently for so-called equivalent subsystems that are actually Nyquist plots of decoupled subsystems shaped by a selected characteristic locus of the interactions matrix. Local controllers of equivalent subsystems independently tuned for stability and specified feasible performance constitute the decentralized controller guaranteeing specified performance of the full system. Unlike standard robust approaches, the proposed technique considers full mean parameter value nominal model, thus reducing conservatism of resulting robust stability conditions. In the context of robust decentralized controller design, the Equivalent Subsystems Method (Kozakova et. al., 2009b) is applied to design a decentralized controller for the nominal model Go(s) as depicted in Fig. 4. G (s) — <J^ Fig. 4. Standard feedback loop under decentralized controller The key idea behind the method is factorisation of the closed-loop characteristic polynomial detF(s) in terms of the split nominal system (60) under the decentralized controller (62) (existence of R _1 (s) is implied by the assumption (62) that det R(s) ^ ) detF(s) = det {I + [G d (s) + G m (s)]R(s) } =det[R- 1 (s) + G d (s) + G m (s)]detR(s) (63) Denote where F 1 (s) = R- 1 (s) + G d (s) + G m (s) = P( S ) + G m (s) (64) P( S ) = R-\ S ) + G d (s) (65) is a diagonal matrix P(s) = diag{p i (s)} mxm . Considering (63) and (64), the stability condition (37b) in Theorem 3.1 modifies as follows N{0, det[P(s) + G m (s)]} + N[0, det R(s)] = n q and a simple manipulation of (65) yields (66) 234 Robust Control, Theory and Applications I + R(s)[G d (s) - P(s)] = I + R(s)G^(s) = (67) where G a He) = diag{G?(8)} mxm =diag{G i {8)-p i {8)} mxm « = 1 m (68) is a diagonal matrix of equivalent subsystems Gf (s) ; on subsystems level, (67) yields m equivalent characteristic polynomials CLCP t eq (s) = 1 + R f (s)Gf (s) f = 1,2,... ,ra (69) Hence, by specifying P(s) it is possible to affect performance of individual subsystems (including stability) through R' 1 (s) . In the context of the independent design philosophy, design parameters p i (s),i = l,2,...,m represent constraints for individual designs. General stability conditions for this case are given in Corollary 3.1. Corollary 3.1 (Kozakova & Vesely, 2009) The closed-loop in Fig. 4 comprising the system (60) and the decentralized controller (62) is stable if and only if 1. there exists a diagonal matrix P(s) = diag{p i {s)} i=1 such that all equivalent subsystems (68) can be stabilized by their related local controllers Rj(s), i.e. all equivalent characteristic polynomials CLCP^{s) = l + R i {s)G e i q {s) , i = l,2,...,m have roots with Re {s} < ; 2. the following two conditions are met VseD : a. det[P(s) + G m (s)]*0 b. N[0,detF(s)] = n q where detP(s) = det(l + G(s)R(s)) and n is the number of open loop poles with Re{s} >0 . In general, p t (s) are to be transfer functions, fulfilling conditions of Corollary 3.1, and the stability condition resulting form the small gain theory; according to it if both P^fs) and G m (s) are stable, the necessary and sufficient closed-loop stability condition is |p(s)- 1 G w (s)| < 1 or <r min [P(s)] > <x max [G m (s)] (71) To provide closed-loop stability of the full system under a decentralized controller, Pi (s), i = l,2,...,m are to be chosen so as to appropriately cope with the interactions G m (s) . A special choice of P(s) is addressed in (Kozakova et al.2009a;b): if considering characteristic functions g z (s)of G m (s) defined according to (35) for i = l,...,m, and choosing P(s) to be diagonal with identical entries equal to any selected characteristic function gk(s) of [-G m (s)], where /ce{l,...,ra} is fixed, i.e. P(s) = -g k (s)I, ke{l,...,m] is fixed (72) then substituting (72) in (70a) and violating the well-posedness condition yields m det[P(s) + G m (s)] = Yl[-g k (s) + gl (s)] = VseD (73) Robust Controller Design: New Approaches in the Time and the Frequency Domains 235 In such a case the full closed-loop system is at the limit of instability with equivalent subsystems generated by the selected g k (s) according to GS(8) = G,(e) + g k (e) .=1,2 m, VseD (74) Similarly, if choosing P(s-a) = -g k (s-a)1 , 0<a<a m where a m denotes the maximum feasible degree of stability for the given plant under the decentralized controller R(s) , then m detF 1 (s-a) = n[-g l (s-ar) + g i (s-o)] = VseD (75) i=l Hence, the closed-loop system is stable and has just poles with Re{s} < -a , i.e. its degree of stability is a . Pertinent equivalent subsystems are generated according to GS(8-a) = G i (8-a) + g k (8-a) i=l,2 m (76) To guarantee stability, the following additional condition has to be satisfied simultaneously m m detF lk =H[-g k (s-a) + g,(s)] = Il r ,k( s )* VseD ( 77 ) Simply put, by suitably choosing a : < a < a m to generate P(s - a) it is possible to guarantee performance under the decentralized controller in terms of the degree of stability a . Lemma 3.1 provides necessary and sufficient stability conditions for the closed- loop in Fig. 4 and conditions for guaranteed performance in terms of the degree of stability. Definition 3.1 (Proper characteristic locus) The characteristic locus g k (s - a) of G m (s - a) , where fixed k e {l,...,m} and a > , is called proper characteristic locus if it satisfies conditions (73), (75) and (77). The set of all proper characteristic loci of a plant is denoted P s . Lemma 3.1 The closed-loop in Fig. 4 comprising the system (60) and the decentralized controller (62) is stable if and only if the following conditions are satisfied \/seD , a > and fixed k g {!,..., m} : 1. g k (s-a)eP s 2. all equivalent characteristic polynomials (69) have roots with Res < -a ; 3. N[0,detF(s-a)] = n qa where F(s -a) = I + G(s - a)R(s - a) ; n is the number of open loop poles with Re{s} > -a . Lemma 3.1 shows that local controllers independently tuned for stability and a specified (feasible) degree of stability of equivalent subsystems constitute the decentralized controller guaranteeing the same degree of stability for the full system. The design technique resulting from Corollary 3.1 enables to design local controllers of equivalent subsystems using any SISO frequency-domain design method, e.g. the Neymark D-partition method (Kozakova et al. 2009b), standard Bode diagram design etc. If considering other performance measures in the ESM, the design proceeds according to Corollary 3.1 with P(s) and G e ik q (s) = G i (s) + g k (s),i = l,2,...,m generated according to (72) and (74), respectively. 236 Robust Control, Theory and Applications According to the latest results, guaranteed performance in terms of maximum overshoot is achieved by applying Bode diagram design for specified phase margin in equivalent subsystems. This approach is addressed in the next subsection. 3.3 Robust decentralized controller design The presented frequency domain robust decentralized controller design technique is applicable for uncertain systems described as a set of transfer function matrices. The basic steps are: 1. Modelling the uncertain system This step includes choice of the nominal model and modelling uncertainty using any unstructured uncertainty (41) -(46) or (55). The nominal model can be calculated either as the mean value parameter model (Skogestad & Postlethwaite, 2005), or the "affine" model, obtained within the procedure for calculating the affine-type additive uncertainty (Kozakova & Vesely, 2007; 2008). Unlike the standard robust approach to decentralized control design which considers diagonal model as the nominal one (interactions are included in the uncertainty), the ESM method applied in the design for nominal performance allows to consider the full nominal model. 2. Guaranteeing nominal stability and performance The ESM method is used to design a decentralized controller (62) guaranteeing stability and specified performance of the nominal model (nominal stability, nominal performance). 3. Guaranteeing robust stability In addition to nominal performance, the decentralized controller has to guarantee closed- loop stability over the whole operating range of the plant specified by the chosen uncertainty description (robust stability). Robust stability is examined by means of the M-A stability condition (47) or the M a f--Q stability condition (59) in case of the affine type additive uncertainty (55). Corollary 3.2 (Robust stability conditions under DC) The closed-loop in Fig. 3 comprising the uncertain system given as a set of transfer function matrices and described by any type of unstructured uncertainty (41) - (46) or (55) with nominal model fulfilling (60), and the decentralized controller (62) is stable over the pertinent uncertainty region if any of the following conditions hold 1. for any (41)-(46), conditions of Corollary 3.1 and (47) are simultaneously satisfied where M(s) = £ k M k (s), k = a,i,o,ia,ii,io and M\ given by (48)-(53) respectively. 2. for (55), conditions of Corollary 3.1 and (59) are simultaneously satisfied. Based on Corollary 3.2, two approaches to the robust decentralized control design have been developed: the two-stage and the direct approaches. 1. The two stage robust decentralized controller design approach based on the M-A structure stability conditions (Kozakova & Vesely, 2008;, Kozakova & Vesely, 2009; Kozakova et al. 2009a). In the first stage, the decentralized controller for the nominal system is designed using ESM, afterwards, fulfilment of the M-A or M a f-Q stability conditions (47) or (59), respectively is examined; if satisfied, the design procedure stops, otherwise the second stage follows: either controller parameters are additionally modified to satisfy robust stability conditions in the tightest possible way (Kozakova et al. 2009a), or the redesign is carried out with modified performance requirements (Kozakova & Vesely, 2009). Robust Controller Design: New Approaches in the Time and the Frequency Domains 237 2. Direct decentralized controller design for robust stability and nominal performance By direct integration of the robust stability condition (47) or (59) in the ESM, local controllers of equivalent subsystems are designed with regard to robust stability. Performance specification for the full system in terms of the maximum peak of the complementary sensitivity M T corresponding to maximum overshoot in individual equivalent subsystems is translated into lower bounds for their phase margins according to (78) (Skogestad & Postlethwaite, 2005) PM > 2 arcsin y2M T j >^-[rad] (78) where PM is the phase margin, Mr is the maximum peak of the complementary sensitivity T(s) = G(s)R(s)[I + G(s)R(s)]- 1 (79) As for MIMO systems M T =a max (T) (80) the upper bound for Mr can be obtained using the singular value properties in manipulations of the M-A condition (47) considering (48)-(53), or the M a f- Q condition (58) considering (57) and (59). The following upper bounds o- max [T (ja>)] for the nominal complementary sensitivity T (s) = G (s)R(s)[I + G (s)R(s)] _1 have been derived: °max[ T o(.H]< °" mi . n[Go ^ ] =L A {a>) \fco additive uncertainty (81) cr max [T (jco)] < r = L K (co), k = i, o, \/co multiplicative input/output uncertainty (82) \£ k (a>)\ a max [T (jcD)] < 1 gmin[gg(M! = L AF {co) Va additive affine-type uncertainty (83) Using (80) and (78) the upper bounds for the complementary sensitivity of the nominal system (81) -(83) can be directly implemented in the ESM due to the fact that performance achieved in equivalent subsystems is simultaneously guaranteed for the full system. The main benefit of this approach is the possibility to specify maximum overshoot in the full system guaranteeing robust stability in terms of cr mSLX (T ) , translate it into minimum phase margin of equivalent subsystems and design local controllers independently for individual single input - single output equivalent subsystems. The design procedure is illustrated in the next subsection. 3.4 Example Consider a laboratory plant consisting of two interconnected DC motors, where each armature voltage (Mi, Mi) affects rotor speeds of both motors (oe>i, oo 2 ). The plant was identified in three operating points, and is given as a set 77 = {G 1 (s),G 2 (s),G 3 (s)} where 238 Robust Control, Theory and Applications G 1 (s) -0.402s + 2.690 s 2 + 2.870s + 1.840 0.003s -0.720 0.006s -1.680 + 11.570s + 3.780 -0.170s + 1.630 - 9.850s + 1.764 s 2 + 1.545s + 0.985 G 2 (s)- -0.342s + 2.290 s 2 + 2.070s + 1.840 0.003s -0.580 0.005s -1.510 s 2 + 10.570s + 3.780 -0.160s + 1.530 s z + 8.850s + 1.764 s z + 1.045s + 0.985 G 3 (s) = -0.423s + 2.830 0.006s -1.930 s 2 + 4.870s + 1.840 s 2 + 13.570s + 3.780 0.004s -0.790 -0.200s + 1.950 s 2 + 10.850s + 1.764 s 2 + 1.945s + 0.985 In calculating the affine nominal model Go(s), all possible allocations of Gi(s), Gi(s), Gs(s) into the 2 2 = 4 polytope vertices were examined (24 combinations) yielding 24 affine nominal model candidates and related transfer functions matrices G^(s) needed to complete the description of the uncertainty region. The selected affine nominal model Go(s) is the one guaranteeing the smallest additive uncertainty calculated according to (41): G (s) = -0.413 s +2.759 s 2 + 3.870s + 1.840 0.004s -0.757 -0.006s -1.807 s 2 + 12.570s + 3.780 -0.187s + 1.791 s z + 10.350s + 1.764 s z + 1.745s + 0.985 The upper bound L AF (co) for To(s) calculated according to (82) is plotted in Fig. 5. Its worst (minimum value) M T = minL Af (co) = 1.556 corresponds to PM > 37.48° according to (78). 3.5 Fig. 5. Plot of Laf(co) calculated according to (82) The Bode diagram design of local controllers for guaranteed PM was carried out for equivalent subsystems generated according to (74) using characteristic locus gi(s) of the matrix of interactions G m (s), i.e. G e i l(s) = G i (s) + g 2 (s) z = l,2. Bode diagrams of equivalent Robust Controller Design: New Approaches in the Time and the Frequency Domains 239 subsystems G^(s),G2i(s) are in Fig. 6. Applying the PI controller design from Bode diagram for required phase margin PM = 39° has yielded the following local controllers R a (s) = 3.367 s +1.27 R 2 (s) = 1.803s + 0.491 Bode diagrams of compensated equivalent subsystems in Fig. 8 prove the achieved phase margin. Robust stability was verified using the original M a /-Q condition (59) with p=2 and qo=l', as depicted in Fig. 8, the closed loop under the designed controller is robustly stable. CO S -20 1 " 40 -fin ~~ "-v ■v ■N X N \ " \ ■V 1 \ *-s v <> > ^ U — 100 omega [rad/s] Fig. 6. Bode diagrams of equivalent subsystems Gl\(s) (left), G%[(s) (right) II II 100 2. 50 1 o -50 - — . - ~-—~ - ~ ■ — .. *^ - ■<■* — - -> «^ - ■^ - ^ 10 10 co [rad/sec] 10" 10 u © [rad/sec] 10- 10 l co [rad/sec] Fig. 7. Bode diagrams of equivalent subsystems GH(s) (left), G%[(s) (right) under designed local controllers Ri(s), R2(s), respectively. 240 Robust Control, Theory and Applications Fig. 8. Verification of robust stability using condition (59) in the form <r max (M a r) < ^ 4. Conclusion The chapter reviews recent results on robust controller design for linear uncertain systems applicable also for decentralized control design. In the first part of the chapter the new robust PID controller design method based on LMP is proposed for uncertain linear system. The important feature of this PID design approach is that the derivative term appears in such form that enables to consider the model uncertainties. The guaranteed cost control is proposed with a new quadratic cost function including the derivative term for state vector as a tool to influence the overshoot and response rate. In the second part of the chapter a novel frequency-domain approach to the decentralized controller design for guaranteed performance is proposed. Its principle consists in including plant interactions in individual subsystems through their characteristic functions, thus yielding a diagonal system of equivalent subsystems. Local controllers of equivalent subsystems independently tuned for specified performance constitute the decentralized controller guaranteeing the same performance for the full system. The proposed approach allows direct integration of robust stability condition in the design of local controllers of equivalent subsystems. Theoretical results are supported with results obtained by solving some examples. 5. Acknowledgment This research work has been supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic, Grant No. 1/0544/09. 6. References Blondel, V. & Tsitsiklis, J.N. (1997). NP-hardness of some linear control design problems. SIAM J. Control Optim., Vol. 35, 2118-2127. Boyd, S.; El Ghaoui, L.; Feron, E. & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Philadelphia. Robust Controller Design: New Approaches in the Time and the Frequency Domains 241 Crusius, C.A.R. & Trofino, A. (1999). LMI Conditions for Output Feedback Control Problems. IEEE Trans. Aut. Control Vol. 44, 1053-1057. de Oliveira, M.C.; Bernussou, J. & Geromel, J.C. (1999). A new discrete-time robust stability condition. Systems and Control Letters, Vol. 37, 261-265. de Oliveira, M.C.; Camino, J.F. & Skelton, R.E. (2000). A convexifying algorithm for the design of structured linear controllers, Proc. 39 nd IEEE CDC, pp. 2781-2786, Sydney, Australia, 2000. Ming Ge; Min-Sen Chiu & Qing-Guo Wang (2002). Robust PID controller design via LMI approach. Journal of Process Control, Vol.12, 3-13. Grman, L. ; Rosinova, D. ; Kozakova, A. & Vesely, V. (2005). Robust stability conditions for polytopic systems. International Journal of Systems Science, Vol. 36, No. 15, 961-973, ISSN 1464-5319 (electronic) 0020-7721 (paper) Gyurkovics, E. & Takacs, T. (2000). Stabilisation of discrete-time interconnected systems under control constraints. IEE Proceedings - Control Theory and Applications, Vol. 147, No. 2, 137-144 Han, J. & Skelton, R.E. (2003). An LMI optimization approach for structured linear controllers, Proc. 42** IEEE CDC, 5143-5148, Hawaii, USA, 2003 Henrion, D.; Arzelier, D. & Peaucelle, D. (2002). Positive polynomial matrices and improved LMI robustness conditions. 15 th IF AC World Congress, CD-ROM, Barcelona, Spain, 2002 Kozakova, A. & Vesely, V. (2007). Robust decentralized controller design for systems with additive af fine-type uncertainty. Int. J. of Innovative Computing, Information and Control (IJICIC), Vol. 3, No. 5 (2007), 1109-1120, ISSN 1349-4198. Kozakova, A. & Vesely, V. (2008). Robust MIMO PID controller design using additive affine-type uncertainty. Journal of Electrical Engineering, Vol. 59, No.5 (2008), 241- 247, ISSN 1335 - 3632 Kozakova, A., Vesely, V. (2009). Design of robust decentralized controllers using the M-A structure robust stability conditions. Int. Journal of Systems Science, Vol. 40, No.5 (2009), 497-505, ISSN 1464-5319 (electronic) 0020-7721 (paper). Kozakova, A.; Vesely, V. & Osusky, J. (2009a). A new Nyquist-based technique for tuning robust decentralized controllers, Kybernetika, Vol. 45, No.l (2009), 63-83, ISSN 0023- 5954. Kozakova, A.; Vesely, V. Osusky, J. (2009b). Decentralized Controllers Design for Performance: Equivalent Subsystems Method, Proceedings of the European Control Conference, ECC09, 2295-2300, ISBN 978-963-311-369-1, Budapest, Hungary August 2009, EUCA Budapest. Peaucelle, D.; Arzelier, D.; Bachelier, O. & Bernussou, J. (2000). A new robust D-stability condition for real convex polytopic uncertainty. Systems and Control Letters, Vol. 40, 21-30 Rosinova, D.; Vesely, V. & Kucera, V. (2003). A necessary and sufficient condition for static output feedback stabilizability of linear discrete-time systems. Kybernetika, Vol. 39, 447-459 Rosinova, D. & Vesely, V. (2003). Robust output feedback design of discrete-time systems - linear matrix inequality methods. Proceedings 2 th IF AC Conf. CSD'03 (CD-ROM), Bratislava, Slovakia, 2003 242 Robust Control, Theory and Applications Skelton, R.E.; Iwasaki, T. & Grigoriadis, K. (1998). A Unified Algebraic Approach to Linear Control Design, Taylor and Francis, Ltd, London, UK Skogestad, S. & Postlethwaite, I. (2005). Multivariable fedback control: analysis and design, John Wiley & Sons Ltd., ISBN -13978-0-470-01167-6 (H/B), The Atrium, Southern Gate. Chichester, West Sussex, UK Vesely, V. (2003). Robust output feedback synthesis: LMI Approach, Proceedings 2 th IF AC Conference CSD'03 (CD-ROM), Bratislava, Slovakia, 2003 Zheng Feng; Qing-Guo Wang & Tong Heng Lee (2002). On the design of multivariable PID controllers via LMI approach. Automatica, Vol. 38, 517-526 11 Robust Stabilization and Discretized PID Control Yoshifumi Okuyama Tottori University, Emeritus Japan 1. Introduction At present, almost all feedback control systems are realized using discretized (discrete-time and discrete-value, i.e., digital) signals. However, the analysis and design of discretized /quantized control systems has not been entirely elucidated. The first attempt to elucidate the problem was described in a paper by Kalman (1) in 1956. Since then, many researchers have studied this problem, particularly the aspect of understanding and mitigating the quantization effects in quantized feedback control, e.g.,(2-4). However, few results have been obtained for the stability analysis of the nonlinear discrete-time feedback system. This article describes the robust stability analysis of discrete-time and discrete-value control systems and presents a method for designing (stabilizing) PID control for nonlinear discretized systems. The PID control scheme has been widely used in practice and theory thus far irrespective of whether it is continuous or discrete in time (5; 6) since it is a basic feedback control technique. In the previous study (7-9), a robust stability condition for nonlinear discretized control systems that accompany discretizing units (quantizers) at equal spaces was examined in a frequency domain. It was assumed that the discretization is executed at the input and output sides of a nonlinear continuous elemet (sensor /actuator) and that the sampling period is chosen such that the size is suitable for discretization in the space. This paper presents a designing problem for discretized control systems on a grid pattern in the time and controller variables space. In this study, the concept of modified Nyquist and Nichols diagrams for nonlinear control systems given in (10; 11) is applied to the designing procedure in the frequency domain. Fig. 1. Nonlinear discretized PID control system. 244 Robust Control, Theory and Applications 2. Discretized control system The discretized control system in question is represented by a sampled-data (discrete-time) feedback system as shown in Fig. 1. In the figure, G(z) is the z-transform of continuous plant G(s) together with the zero-order hold, C(z) is the z-transform of the digital PID controller, and V\ and V^ are the discretizing units at the input and output sides of the nonlinear element, respectively The relationship between e and w + = N^(e) is a stepwise nonlinear characteristic on an integer-grid pattern. Figure 2 (a) shows an example of discretized sigmoid-type nonlinear characteristic. For C-language expression, the input /output characteristic can be written as e f = 7 * (double) (int)(e/ 7) u = 0.4 * e f + 3.0 * atan(0.6 * e f ) (1) u f = 7 * (double) (int)(w/7), where (int) and (double) denote the conversion into an integral number (a round-down discretization) and the reconversion into a double-precision real number, respectively. Note that even if the continuous characteristic is linear, the input /output characterisitc becomes nonlinear on a grid pattern as shown in Fig. 2 (b), where the linear continuous characteristic is chosen asu = 0.85 * e + . In this study, a round-down discretization, which is usually executed on a computer, is applied. Therefore, the relationship between e + and w + is indicated by small circles on the stepwise nonlinear characteristic. Here, each signal e + , w + , • • • can be assigned to an integer number as follows: e + e{... , -37, -27, -7,0, 7, 27, 37, • ■ ■•}, « + €{••• ,-37,-27,-7,0, 7, 27, 37,- ••}. where 7 is the resolution of each variable. Without loss of generality, hereafter, it is assumed that 7 = 1.0. That is, the variables e + , w + , • • • are defined by integers as follows: e f ,u f eZ, Z= { 3,-2, -1,0, 1,2,3,- ••}. On the other hand, the time variable t is given as t £ {0, h, 2/z, 3h, • • •} for the sampling period h. When assuming h = 1.0, the following expression can be defined: tez +f Z+ = {0,1,2,3,- ••}. Therefore, each signal e^it), u*(t), • • • traces on a grid pattern that is composed of integers in the time and controller variables space. The discretized nonlinear characteristic u f = N d (e f ) =Ke f + g(e f ), < K < 00, (2) as shown in Fig. 2(a) is partitioned into the following two sections: \g(e*)\<g<™, (3) for \e f \ < £, and \g(e*)\<p\e*\, 0<p<K, (4) Robust Stabilization and Discretized PID Control 245 ft '(V ) J\£lL . X '-''" J J X* '- l s _ -' -* V - -'■'X- /'> -J ii ■i e -f n fl -' - '> ^" - ■^ ,* -' y J s^- J X / (a) (b) Fig. 2. Discretized nonlinear characteristics on a grid pattern. for |e + | > e. (In Fig. 2 (a) and (b), the threshold is chosen as £ = 2.0.) Equation (3) represents a bounded nonlinear characteristic that exists in a finite region. On the other hand, equation (4) represents a sectorial nonlinearity for which the equivalent linear gain exists in a limited range. It can also be expressed as follows: < g(e r y < pe tz < Ke t2 J2 (5) When considering the robust stability in a global sense, it is sufficient to consider the nonlinear term (4) for |e + | > £ because the nonlinear term (3) can be treated as a disturbance signal. (In the stability problem, a fluctuation or an offset of error is assumed to be allowable in |e + 1 < e.) "1 1 + qS £*(') + + -j 8(e) Fig. 3. Nonlinear subsystem g(e). -) e * **(•) P* J y' UTfa n *r\ ■ ^ ( + J " L Fig. 4. Equivalent feedback system. 246 Robust Control, Theory and Applications 3. Equivalent discrete-time system In this study, the following new sequences e^ (k) and v^ (k) are defined based on the above consideration: e«(k)=e$ l (k)+q-^-, (6) „;+(*) = *+,(*) -ft. ^!M, (7) where q is a non-negative number, eJi(fc) and »J,();) are neutral points of sequences e + (fc) and iit) _ ,* w+ . >,>-,) _ (8) ,i,w - sMta, (9) and Ae + (/c) is the backward difference of sequence e f (k), that is, Ae\k) =e\k)-e\k-l). (10) The relationship between equations (6) and (7) with respect to the continuous values is shown by the block diagram in Fig. 3. In this figure, S is defined as *v> - r tt£- (11) Thus, the loop transfer function from z?* to e* can be given by W(f},q,z), as shown in Fig. 4, where (l + ^(z))G(z)C(z) WUM/ZJ i + (K + ^( z ))G(z)C(z)' U) and r 7 , d' are transformed exogenous inputs. Here, the variables such as v*, u' and y' written in Fig. 4 indicate the z-transformed ones. In this study, the following assumption is provided on the basis of the relatively fast sampling and the slow response of the controlled system. [Assumption] The absolute value of the backward difference of sequence e(k) does not exceed 7, i.e., \Ae(k)\ = \e(k)-e(k-l)\< 7 . (13) If condition (13) is satisfied, Ae*(k) is exactly ±7 or because of the discretization. That is, the absolute value of the backward difference can be given as \Ae\k)\ = \e\k) - e\k - 1)\ = 7 or 0. □ The assumption stated above will be satisfied by the following examples. The phase trace of backward difference Ae + is shown in the figures. Robust Stabilization and Discretized PID Control 247 Fig. 5. Nonlinear characteristics and discretized outputs. 4. Norm inequalities In this section, some lemmas with respect to an £2 norm of the sequences are presented. Here, we define a new nonlinear function f(e)~g(e) + fle. (14) When considering the discretized output of the nonlinear characteristic, z? + = g(e f ), the following expression can be given: f(e\k))=v i (k)+lie\k). (15) From inequality (4), it can be seen that the function (15) belongs to the first and third quadrants. Figure 5 shows an example of the continuous nonlinear characteristics u = N(e) and f(e), the discretized outputs u f = N^(e f ) and f(e f ), and the sector (4) to be considered. When considering the equivalent linear characteristic, the following inequality can be defined: < 1K*) := =^p < 2/5. (16) When this type of nonlinearity ip(k) is used, inequality (4) can be expressed as S(k)=g(e*(k)) = (ip(k)-ll)e\k). (17) For the neutral points of e*(k) and v^{k), the following expression is given from (15): \{f(e\k))+f{e\k-l)))=vl l {k)+^l l (k). (18) Moreover, equation (17) is rewritten as v^ m (k) = (ip(k) — /3)ejj(fc). Since |ej,(fc)| < \e m (k)\, the following inequality is satisfied when a round-down discretization is executed: \vt(k)\ < fS\el(k)\ < P\e m (k)\. (19) 248 Robust Control, Theory and Applications Based on the above premise, the following norm conditions are examined [Lemma-1] The following inequality holds for a positive integer p: \\vi(k)\\2.p < P\\em(k)h P < P\\e m (k)\\ 2 , p . (20) Here, || • \\ 2/ p denotes the Euclidean norm, which can be defined by f p \ 1/2 Mk)h P :=[E^( k ) \k=l (Proof) The proof is clear from inequality (19). D [Lemma-2] If the following inequality is satisfied with respect to the inner product of the neutral points of (15) and the backward difference: {vl,(k)+llel I (k),Ae*(k)) p >0, (21) the following inequality can be obtained: \KKk)h, v <f>\\e^{k)h, v (22) for any q > 0. Here, (-r)p denotes the inner product, which is defined as (xi(k),x 2 (k)) p =£x 1 (k)x 2 (k). fc=i (Proof) The following equation is obtained from (6) and (7): 2 ll#(*)lli P - Wtfmlp = 2 ll4,(*)lll P - \W m {k)f %v + *& • (vUk) + ^ m (k)Ae\k)) p . (23) Thus, (22) is satisfied by using the left inequality of (20). Moreover, as for the input of g* (•), the following inequality can be obtained from (23) and the right inequality (20): \K\m2, P < P\K(k)h P . (24) □ The left side of inequality (21) can be expressed as a sum of trapezoidal areas. [Lemma-3] For any step p, the following equation is satisfied: <V) := {vl(k)+ftel{k)M(k) ) v = \ E(f(e f (k))+f(e\k-l)))Ae\k). (25) z k=i (Proof) The proof is clear from (18). □ In order to understand easily, an example of the sequences of continuous /discretized signals and the sum of trapezoidal areas is depicted in Fig. 6. The curve e and the sequence of circles e + show the input of the nonlinear element and its discretized signal. The curve u and the sequence of circles w + show the corresponding output of the nonlinear characteristic and its discretized signal, respectively. As is shown in the figure, the sequences of circles e + and u f trace on a grid pattern that is composed of integers. The sequence of circles v f shows Robust Stabilization and Discretized PID Control 249 p ieUflff /I \, /'I Jl\ / ^vL /T^w -^m i f_J° ° ^J**-— ^^C* * X\ J *V_ ' 20 '\ . o^Jftt"^^^^. 1,1114 f '' '' $Kn*n SSSJy 0/ t \ or i . 5 . 'u \ iii::::::::p 1 u 1 y Re) k^"""~ "" in 'f(^) ' ! 7 t V II III II III rill Fig. 6. Discretized input/output signals of a nonlinear element. the discretized output of the nonlinear characteristic g(-). The curve of shifted nonlinear characteristic f(e) and the sequence of circles f(e f ) are also shown in the figure. In general, the sum of trapezoidal areas holds the following property. [Lemma-4] If inequality (13) is satisfied with respect to the discretization of the control system, the sum of trapezoidal areas becomes non-negative for any p, that is, a(p) > 0. (26) (Proof) Since f(e f (k)) belongs to the first and third quadrants, the area of each trapezoid r(k) :=^(/(e + W) + /(e + (fc-l)))A £ + (fc) (27) On the other hand, the trapezoidal area r(k) is non-positive when e(k) decreases (increases) in the first (third) quadrant. Strictly speaking, when (e{k) > and Ae(k) > 0) or (e(k) < and Ae(k) < 0), r(k) is non-negative for any k. On the other hand, when (e(k) > and Ae(k) < 0) or (e(k) < and Ae(k) > 0), r(k) is non-positive for any k. Here, Ae(k) > corresponds to Ae f (k) = j or (and Ae(k) < corresponds to Ae f (k) = —7 or 0) for the discretized signal, when inequality (13) is satisfied. The sum of trapezoidal area is given from (25) as: a{p) E T (*)- k=l (28) Therefore, the following result is derived based on the above. The sum of trapezoidal areas becomes non-negative, a{p) > 0, regardless of whether e(k) (and e + (k)) increases or decreases. Since the discretized output traces the same points on the stepwise nonlinear characteristic, the sum of trapezoidal areas is canceled when e(k) (and e*(k) decreases (increases) from a certain point (e f (k),f(e f (k) ) ) in the first (third) quadrant. (Here, without loss of generality, the response of discretized point (e + (k) , / (e f (k))) is assumed to commence at the origin.) Thus, the proof is concluded. □ 250 Robust Control, Theory and Applications 5. Robust stability in a global sense By applying a small gain theorem to the loop transfer characteristic (12), the following robust stability condition of the discretized nonlinear control system can be derived [Theorem] If there exists a q > in which the sector parameter f> with respect to nonlinear term g(-) satisfies the following inequality, the discrete-time control system with sector nonlinearity (4) is robust stable in an £2 sense: B < Bq = K • t](qo,coo) = max min K • rj(q,cv), (29) q co when the linearized system with nominal gain K is stable. The //-function is written as follows: -qCl sin + \/q 2 CL 2 sin 2 6 + p 2 + 2p cos 6 + 1 *l(q,oo) := , Vo; £ [Q,oo c ], (30) where Cl(co) is the distorted frequency of angular frequency co and is given by <5(e^)=;0(o;)=;|tan^) / ; = v^T (31) and co c is a cut-off frequency In addition, p(co) and 6 (to) are the absolute value and the phase angle of KG(e^)C(e^), respectively (Proof) Based on the loop characteristic in Fig. 4, the following inequality can be given with respect to z = e^ h : Ikm(^)ll2,p < ^11^(^)112^ + ^11^(^)112^ + SUp|W(^,^,z)| - ||^^ + (z)||2,p- (32) z=l Here, r' m (z) and d' m (z) denote the z-transformation for the neutral points of sequences r' (k) and d'(k), respectively. Moreover, C\ and c^ are positive constants. By applying inequality (24), the following expression is obtained: l-/5.sup|W(/5, (? , 2 )|j||4( 2 )|| 2 , p < Cl ||4(z)|| 2 , p + c 2 ||d;„( 2 )|| 2 , p . (33) Therefore, if the following inequality (i.e., the small gain theorem with respect to £2 gains) is valid, |WQ3,<7,e jcvh\ {l+iqCL{co))P(e^ h )C{e^ h ) (l+jqn(to))p(to)ei e ^ K+ (K + ipqCL(co))p(co)ei e ( w ) 1 1 + (K + iPqO,(to))P{ei wh )C{ei u > h ) (34) the sequences e^ik), e m (k), e(k) and y(k) in the feedback system are restricted in finite values when exogenous inputs r(k), d(k) are finite and p — > 00. (The definition of £2 stable for discrete-time systems was given in (10; 11).) From the square of both sides of inequality (34), fi 2 p 2 (l + q 2 n 2 ) < (K + Kp cos 6-ppqCl sin 6) 2 + (Kp sin. 6 + PpqO, cos 6) 2 . Robust Stabilization and Discretized PID Control 251 Fig. 7. An example of modified Nichols diagram (M = 1.4, c q = 0.0,0.2, • • • ,4.0). Thus, the following quadratic inequality can be obtained: P 2 p 2 < -IfiKpqCt sinO + K 2 (I + p cos e) 2 + K 2 p 2 sin 2 0. Consequently, as a solution of inequality (35), -KqClsm6 + K\/q 2 n 2 sin 2 + p 2 + 2p cos + 1 P < = Kt](cj,cv) (35) (36) □ 6. Modified Nichols diagram In the previous papers,the inverse function was used instead of the ^/-function, i.e., Using the notation, inequality (29) can be rewritten as follows: M = £{qo,co ) = min max g(q,a>) < -. q cv p When q = 0, the ^-function can be expressed as: £(o,«>) Vi0 2 + 2pcose + l \T(ei wh )\, where T(z) is the complementary sensitivity function for the discrete-time system. ft is evident that the following curve on the gain-phase plane, £(0,a>) = M, (M: const.) (37) (38) (39) 252 Robust Control, Theory and Applications corresponds to the contour of the constant M in the Nichols diagram. In this study, since an arbitrary non-negative number q is considered, the ^-function that corresponds to (38) and (39) is given as follows: M. (40) -qCi sin 6 + y q 2 Ci 2 sin 2 6 + p 2 + 2p cos 6 + 1 From this expression, the following quadratic equation can be obtained: (M 2 - l)p 2 + 2pM(Mcos0 - qCLsinO) + M 2 = 0. (41) The solution of this equation is expressed as follows: M ,,, „ _ . rtN , M M 2 - -(Mcosfl - qCLsinO) ± — ^(McosO- ^Qsinfl) 2 - (M 2 - 1). (42) The modified contour in the gain-phase plane (0, p) is drawn based on the equation of (42). Although the distorted frequency Q is a function of w, the term qd = c q > is assumed to be a constant parameter. This assumption for M contours was also discussed in (11). Figure 7 shows an example of the modified Nichols diagram for Cq > and M = 1.4. Here, GPi is a gain-phase curve that touches an M contour at the peak value (M p = g(0,<Vp) = 1.4). On the other hand, GP2 is a gain-phase curve that crosses the = —180° line and all the M contours at the gain crossover point P2. That is, the gain margin gM becomes equal to — 201og 10 M/(M + 1) = 4.68[dB]. The latter case corresponds to the discrete-time system in which Aizerman's conjecture is valid (14; 15). At the continuous saddle point P2, the following equation is satisfied: f d -^l) =0. (43) Evidently, the phase margin p^ is obtained from the phase crossover point Q2. 7. Controller design The PID controller applied in this study is given by the following algorithm: k u c (k) = K v u\k) + Q £ u\j) + QAi/ + (£), (44) where Au f (k) = u f (k) — u f (k — 1) is a backward difference in integer numbers, and each coefficient is defined as K p , Q, Q e Z+, Z+ = {0,1,2,3 • • • }. Here, Kp, Q, and Q correspond to Kp, Kph/Tj, and KpTp/h in the following (discrete-time z-transform expression) PID algorithm: C( 2 )=K p (l + ^^ + ^l-^)). (45) We use algorithm (44) without division because the variables u f , u c , and coefficients Kp, Cj, Q are integers. Robust Stabilization and Discretized PID Control 253 Using the z-transform expression, equation (44) is written as: u c (z) = C(z)u(z) = (K v + Q(l + z" 1 + z" 2 + • • • ) + Q(l - z" 1 )) u{z). In the closed form, controller C(z) can be given as C(z)=K p + C i -—^— T +C d (l-z- 1 ) (46) 1 — z 1 for discrete-time systems. When comparing equations (45) and (46), Q and Q become equal to Kph/Tj and KpTjj/h, respectively. The design method adopted in this paper is based on the classical parameter specifications in the modified Nichols diagram. This method can be conveniently designed, and it is significant in a physical sense (i.e., mechanical vibration and resonance). Furthermore, in this article, PID-D 2 is considered. The algorithm is written as u c (k) = K p u\k) + Q £ u\j) + C dl Au\k) + Q 2 AV(/c), 7=0 where A 2 u\k) = Au\k) - Au\k - 1) = u\k) - 2u\k - 1) + u\k - 2). Thus, the controller C(z) can be given as C(z) = X p + Q. r -^ T + Q 1 (l-z- 1 ) + Q 2 (l-2z- 1 +z- 2 ) for discrete-time systems. 8. Numerical examples [Example-l] Consider the following third order controlled system: G(s) where K x = 0.0002 = 2.0 x 10" 4 . (s + 0.04) (s + 0.2) (s + 0.4)' K p Q Q £o gM[dB] PM[deg] M p (i) 100 7.72 34.9 1.82 (ii) 100 3 0.98 5.92 23.8 2.61 (iii) 100 3 120 11.1 35.4 1.69 (iv) 50 10.8 48.6 1.29 (v) 50 2 1.00 7.92 30.6 1.99 (vi) 50 2 60 13.3 40.5 1.45 (47) (48) (49) Table 1. PID parameters for Example-l (gM- gain margins, Pm : phase margins, M p : peak values, /5q: allowable sectors). 254 Robust Control, Theory and Applications p / GP 2 /7 ^^^^^ llilK^ Wua % ^li^iw - -k /s^ -135 $ / ;/l0 ^^^^^ =j0^/ Wm0^ Fig. 8. Modified contours and gain-phase curves for Example-1 (M = 1.69, 0.0,0.2, ,4.0). The discretized nonlinear characteristic (discretized sigmoid, i.e. arc tangent (12)) is as shown in Fig. ?? (a). In this article, the resolution value and the sampling period are assumed to be 7 = 1.0 and h = 1.0 as described in section 2. When choosing the nominal gain K = 1.0 and the threshold e = 2.0, the sectorial area of the discretized nonlinear characteristic for e < \e\ can be determined as [0.5, 1.5] drawn by dotted lines in the figure. Figure 8 shows gain-phase curves of KG(ei wh )C(ei wh ) on the modified Nichols diagram. Here, GPi, GP2, and GP3 are cases (i), (ii), and (iii), respectively. The PID parameters are specified as shown in Table 1. The gain margins gM, the phase margin p^ and the peak value Mp can be obtained from the gain crossover points P, the phase crossover points Q, and the points of contact with regard to the M contours, respectively. The max-min value j6q is calculated from (29) (e.g., (ii)) as follows: Bq = max min K • rj(q,co) q co K-rj(q 0/ cv ) = 0.98. Therefore, the allowable sector for nonlinear characteristic g(-) is given as [0.0, 1.98]. The stability of discretized control system (ii) (and also systems (i),(iii)) will be guaranteed. In this example, the continuous saddle point (43) appears (i.e., Aizerman's conjecture is satisfied). Thus, the allowable interval of equivalent linear gain K^ can be given as < 1Q < 1.98. In the case of (i) and (iii), /3q becomes not less than K. However, from the definition of (4), f> in the tables should be considered /3q = $ = 1.0. Figure 9 shows step responses for the three cases. In this figure, the time-scale line is drawn in 10/z increments because of avoiding indistinctness. Sequences of the input u f (k) and the output u\ of PID controller are also shown in the figure. Here, mJ(Zc) is drawn to the scale of 1/100. Figure 10 shows phase traces (i.e., sequences of (e(k),Ae(k)) and (e + ((fc), Ae + (k))). As is obvious from Fig. 10, assumtion (13) is satisfied. The step response (i) remains a sutained oscillation and an off-set. However, as for (ii) and (iii) the responses are improved by using the PID, especially integral (I: a summation in this paper) algorithm. The discretized linear characteristic as shown in Fig. ?? (b) is also considered here. In the figure, the sectorial area of the discretized characteristic for e < \e\ can be determined as [0.5, 0.85] drawn by dotted lines, and the nominal gain is given as K = 0.675. When Robust Stabilization and Discretized PID Control 255 i in W 1 J-L )_£ / j"" \* ^ ^ f .^. „ ^ J f \ V \ ^ 4 ^ V/ >^ Vh- / \ v/ ^ ^' r\-J j / Uc u) u uJ n in, 1 in ?nn t -k \ t ^ 1 t Fig. 9. Step responses for Example-1. Fig. 10. Phase traces for Example-1. normalizing the nominal gain for K = 1.0 (i.e., choosing the gain constant K2 = Ki/ 0.675), the sectorial area is determined as [0.74, 1.26]. In this case, an example of step responces is depicted in Fig. 11. The PID parameters used here are also shown in Table 1. [Example-2] Consider the following fourth order controlled system: G(s) (s + 0.04) (s + 0.2) (s + 0.4) (s + 1.0) ' (50) where K\ = 0.0002 = 2.0 x 10~ 4 . The same nonlinear characteristic and the nominal gain are chosen as shown in Example-1. Figure 12 shows gain-phase curves of KG(e^ coh )C(e^ coh ) on the modified Nichols diagram. Here, GPi, GP2, GP3 and GP4 are cases (i), (ii), (iii) and (iv) in Table 2, respectively. In this example, PID-D 2 control scheme is also used. The PID-D 2 parameters are specified as shown 256 Robust Control, Theory and Applications r \ f</rt \'{ M / 7 X "V / / (v / / \ (i\ ') ' V S u UJ jr "4k W LaW I _L 1 J_ - - - n jj in 200 r -* Fig. 11. Step responses for Example-1 (Discretized linear case). p 10 ^GPx §!§§§^^B f^flB^ y\ \ \ \^ \^x s C^ , o^^^^_ -10 ^ -135 e GPyK / / / // / - P ^GP 4 Fig. 12. Modified contours and gain-phase curves for Example-2 (M = 2.14, c q =0.0,0.2,... ,4.0). in the table. The max-min value /3q is calculated from (29) (e.g., (iv)) as follows: 3n = max min K • rj(q,cv) q co K 'tl(q ,tv ) = 0.69. Therefore, the allowable sector for nonlinear characteristic g(-) is given as [0.0, 1.69]. The stability of discretized control system (ii) (and also systems (i),(iii),(iv)) will be guaranteed. In this example, the continuous saddle point (43) appears (i.e., Aizerman's conjecture is satisfied). Thus, the allowable interval of equivalent gain Kg can be given as < Kg < 1.69. As is shown in Fig. 13, the step response (i) remains a sutained oscillation and an off-set. However, as for (ii), (iii) and (iv) the responses are improved by using PI, PID and PID-D 2 algorithm (D 2 : a second difference). Robust Stabilization and Discretized PID Control 257 r \ _jh J . _j \ - ^ " J-0 ti- M ^-A vt (jjj) ^=~ — t~ \tJ-X 3^ IjfcH nA^- T VI \ ¥ ^ / \ /"^'' y ' V-^ --'" \ ^^^"^ st -if A It - °°° "11 p-Hl t atii.^ J° s *<? ,*?■ , *" 4 S . T r J ■ 1" 'a,r to JL ■■ ~£ _iio _2JII] ^t=A i fej" ■"i Fig. 13. Step responses for Example-2. 1° If / G?3 / >^ ^^J /GP 2 /^ »y)v / ^>^^^:^^^^^^k (uu( \^5\. - =i0 / /^/ "135 ^ iff ' f / j / / -10 If / // Fig. 14. Modified contours and gain-phase curves for Example-3 (M = 1.44, ^=0.0, •••,4.0). [Example-3] Consider the following nonminimum phase controlled system: G{s) £ 2 (s + 0.2)(-s + 0.4) (s + 0.02) (s + 0.04) (s + 1.0)' (51) K p Q Qi Q2 fa g M [dB] PMtdeg] M p (i) 80 ¥> 6.8 37.2 1.79 (ii) 80 3 0.69 4.69 20.9 3.10 (iii) 80 3 60 1.00 6.63 27.4 2.26 (iv) 80 3 60 120 P 7.76 28.8 2.14 Table 2. PID-D 2 parameters for Example-2. 258 Robust Control, Theory and Applications u \ ' s A / \\ v( *) p ^i iiii ) / ^ ^ i v \ / ^ SI!," s v. V li \ s t _\ ^ ■^ V : 1 7 M ■^ *'. r / = / ij c'O - *> j" ■ . ^ X / u/ / ■„ u( J i' ■^ inr ZOO t-k * --=q "\ % . ■* Fig. 15. Step responses for Example-3. where K3 = 0.001 = 1.0 x 10 ~ 3 . Also, in this example, the same nonlinear characteristic and the nominal gain are chosen as shown in Example-1. The modified Nichols diagram with gain-phase curves of KG(e^ coh )C(eJ coh ) is as shown in Fig. 14. Here, GPi, GP2 and GP3 are cases (i), (ii), and (iii), and the PID parameters are specified as shown in Table 3. Figure 15 shows time responses for the three cases. For example, in the case of (iii), although the allowable sector of equivalent linear gain is < Kg < 5.9, the allowable sector for nonlinear characteristic becomes [0.0, 1.44] as shown in Table 3. Since the sectorial area of the discretized nonlinear characteristic is [0.5, 1.5], the stability of the nonlinear control system cannot be guaranteed. The response for (iii) actually fluctuates as shown in Figs. 15 and 16. This is a counter example for Aizerman's conjecture. 9. Conclusion In this article, we have described robust stabilization and discretized PID control for continuous plants on a grid pattern with respect to controller variables and time elapsed. A robust stability condition for nonlinear discretized feedback systems was presented along with a method for designing PID control. The design procedure employs the modified Nichols diagram and its parameter specifications. The stability margins of the control system are specified directly in the diagram. Further, the numerical examples showed that the time responses can be stabilized for the required performance. The concept described in this article will be applicable to digital and discrete-event control system in general. K p Q Q fa g M [dB] PM[deg] M p (i) 100 0.92 15.5 40.6 1.44 (ii) 100 2 0.71 14.7 27.7 2.09 (iii) 100 4 40 0.44 15.3 18.1 3.18 Table 3. PID parameters for Example-3. Robust Stabilization and Discretized PID Control 259 1 Ae r-TTV PrpRt e vl r \ / AH 7 rf K WWJliJ l Fig. 16. Phase traces for Example-3. 10. References [1] R. E. Kalman, "Nonlinear Aspects of Sampled-Data Control Systems", Proc. of the Symposium on Nonlinear Circuit Analysis, vol. VI, pp.273-313, 1956. [2] R. E. Curry, Estimation and Control with Quantized Measurements, Cambridge, MIT Press, 1970. [3] D. F. Delchamps, "Stabilizing a Linear System with Quantized State Feedback", IEEE Trans, on Automatic Control, vol. 35, pp. 916-924, 1990. [4] M. Fu, "Robust Stabilization of Linear Uncertain Systems via Quantized Feedback", Proc. of IEEE Int. Conf. on Decision and Control, TuA06-5, 2003. [5] A. Datta, M.T. Ho and S.P Bhattacharyya, Structure and Synthesis of PID Controllers, Springer- Verlag, 2000. [6] F. Takemori and Y. Okuyama, "Discrete-Time Model Reference Feedback and PID Control for Interval Plants" Digital Control 2000:Past, Present and Future of PID Control, Pergamon Press, pp. 260-265, 2000. [7] Y. Okuyama, "Robust Stability Analysis for Discretized Nonlinear Control Systems in a Global Sense", Proc. of the 2006 American Control Conference, Minneapolis, USA, pp. 2321-2326, 2006. [8] Y Okuyama, "Robust Stabilization and PID Control for Nonlinear Discretized Systems on a Grid Pattern", Proc. of the 2008 American Control Conference, Seattle, USA, pp. 4746-4751, 2008. [9] Y Okuyama, "Discretized PID Control and Robust Stabilization for Continuous Plants", Proc. of the 17th IFAC World Congress, Seoul, Korea, pp. 1492-1498, 2008. [10] Y Okuyama et al., "Robust Stability Evaluation for Sampled-Data Control Systems with a Sector Nonlinearity in a Gain-Phase Plane" Int. J. of Robust and Nonlinear Control, Vol. 9, No. 1, pp. 15-32, 1999. [11] Y Okuyama et al., "Robust Stability Analysis for Non-Linear Sampled-Data Control Systems in a Frequency Domain", European Journal of Control, Vol. 8, No. 2, pp. 99-108, 2002. [12] Y Okuyama et al., "Amplitude Dependent Analysis and Stabilization for Nonlinear Sampled-Data Control Systems", Proc. of the 15th IFAC World Congress, T-Tu-M08, 2002. 260 Robust Control, Theory and Applications [13] Y. Okuyama, "Robust Stabilization and for Discretized PID Control Systems with Transmission Delay", Proc. of IEEE Int. Conf. on Decision and Control, Shanghai, P. R. China, pp. 5120-5126, 2009. [14] L. T. Grujic, "On Absolute Stability and the Aizerman Conjecture", Automatica, pp. 335-349. 1981. [15] Y. Okuyama et ah, "Robust Stability Analysis for Nonlinear Sampled-Data Control Systems and the Aizerman Conjecture", Proc. of IEEE Int. Conf. on Decision and Control, Tampa, USA, pp. 849-852, 1998. 12 Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error Makoto Katoh Osaka Institute of Technology Japan 1. Introduction In this section, the small gain theorem is introduced as a background theory of this chapter. Then, a large mission on safety and a small mission on analytic solutions are introduced after indicating the some problems in discussing robust PI control systems. Moreover, the way how it came to be possible to obtain the analytic solution of PI control adjustment for the concrete robust control problems with uncertain modeling error which is impossible using the space theory for MIMO systems, is shown for a SISO system. The worst lines of closed loop gain margin were shown in a parameter plane. Finally, risk, merit and demerit of the robust control is discussed and the countermeasure for safeness of that is introduced. And some theme, eg., in the lag time system, the MIMO system and a class of non-linear system for expansion of the approach of this chapter is introduced. Many researchers have studied on many kinds of robust system recently. The basic robust stability concept is based on the small gain theorem (Zbou K. with Doyle F. C. and Glover K., 1996). The theorem insists that a closed loop system is internal (robust) stable sufficiently and necessary if the H^ norm of the nominal closed loop transfer function is smaller than the inverse of H^ norm of the any uncertainty of feedback elements. (Fig. 1) Moreover, the expansion of the theorem claims that a closed loop system is stable sufficiently if the product of H^ norms of open loop transfer functions is smaller than 1 when the forward and the feedback transfer functions are both stable. i — ► A( s ) + i W^ AGO ^ ( ^ ^ K W(s) 1 \ . + when IIaII < y if llAll < — then internal stable )< " llo ° " llo ° y Fig. 1. Feed back system configuration with unknown feedback element In MIMO state space models (A,B,C,D), a necessary and sufficient condition using LMI (Linear Matrix Inequality) for the above bounded norm of controlled objects is known as the following Bounded Real Lemma (Zhou K. And Khargonekar P.P., 1988) using the Riccati unequality and Shure complement. 262 Advances in Reinforcement Learning 3P = P* >0 such that i + A'P PB C B T P Y D C D 1 :0o|G(s (0) A gain margin between the critical closed loop gain of a dependent type IP controller by the Furwits criteria and the analytical closed loop gain solution when closed loop Hardy space norm became 1, and the parametric stability margin (Bhattacharyya S. P., Chapellat H., and Keel L. H., 1994; Katoh 2010) on uncertain time constant and damping coefficient were selected in this chapter for its easiness and robustness although it was expected also using this lemma that internal stable concrete conditions for controlled objects and forward controllers may obtain. One of H^ control problems is described to obtain a robust controller K(s) when Hardy space norm of closed loop transfer function matrix is bounded like Fig.2 assuming various (additive, multiplicative, left co-prime factor etc.) uncertainty of controlled objects P(s) (Zbou K. with Doyle F. C. and Glover K., 1996). z = Ow * = P 11 +P 12 K(I-P 22 K) 1 P 21 |®L<7 Fig. 2. Feed back system configuration for obtained robust control K(s) when Hardy space norm of closed loop transfer function matrix is bounded The purpose of this chapter for the robust control problem is to obtain analytical solution of closed loop gain of a dependent type IP controller and analyze robustness by closed loop gain margin for 2 nd order controlled objects with one-order feedback like (left co-prime factor) uncertainty as Fig.l in some tuning regions of IP controller when Hardy space norm of closed loop transfer function matrix is bounded less than 1. Though another basic robust problem is a cooperation design in frequency region between competitive sensitivity and co-sensitivity function, it was omitted in this chapter because a tuning region of IP control was superior for unknown input disturbance other tuning region was superior for unknown reference disturbance. However, there is some one not simple for using higher order controllers with many stable zeros and using the norm with window (Kohonen T., 1995, 1997) for I in Hardy space for evaluating the uncertainty of models. Then, a number of robust PI or PID controller and compensator design methods have recently been proposed. But, they are not considered on the modelling error or parameter uncertainty. Our given large mission is to construct safe robust systems using simple controllers and simple evaluating method of the uncertainty of models. Then, we have proposed robust PI controllers for controlled objects without stable zeros (Katoh M., 2008, 2009). Our small mission in this chapter is to obtain analytical solution of controller gain with flat Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 263 gain curve in a band width as Butter-worse filter for the 3 rd order closed systems with one-order modelling errors and to show the robust property by loop gain margin for damping coefficients of nominal controlled objects and time constants of missing objects (sensor and signal conditioner) using Table Computation Tool (Excel: Microsoft Co. LTD). It is interesting and important historically that infinity time constant is contained in the investing set though it isn't existing actually. Moreover, we confirm the robustness for a parameter change by raising and lowering of step response using CAD Tool (Simulink: Mathworks Co. LTD). Risk of Integral term of PI controller when disconnecting the feedback line can be rescued by M/A station used in many industrial applications or by shutdown of the plant in our standing point. Then, we show a simple soft M/A station for simulation with PI controllers in appendix. This method is not actually because it becomes complicated to computation for higher order objects contained plants with lag time as pointed out in appendix but useful. 2. System description In this section, a description of the higher order generalized system for later 2 nd order examples with one-order modeling error is presented although they may not computed concretely. 2.1 Normalized transfer function In this section, how to normalize and why to normalize transfer functions are explained. The following transfer functions of controlled objects Eq. (1) with multiplicative one-order modeling error Eq. (2) are normalized using a general natural angular frequency co* n and gain K* = K K S as Eq. (3) although the three positions distributed for normalization are different. g(s)=k y\ 2< ™ : — T n (1) w s +2g i m ni s + w ni M s + ctj K. £S + l G l (s) = ^G(s)H(s) (3) es *»<« + p^y^ + - + /K 2r+ ^r< +i y 2r+q Moreover, converting the differential operator s to s as _ A S _ (A) the following normalized open loop transfer function is obtained: Gl ( J ) = —n + l n -n ^"T Where n = 2r+C l ( 5 ) 264 Advances in Reinforcement Learning Neglecting one-order modeling error, the following normalized open loop transfer function is obtained: G{J) = — — ^— where n = 2r + q (6) s +Y n -x s ••• + r^+l 2.2 State space models In this section, 3 kinds of description on normalized state space models are shown although they may not computed concretely. First shows a continuous realization form of the higher order transfer functions to a SISO system. Second shows a normalized sampled system form for the first continuous realization on sampling points. Third shows a normalized continuously approximated form using logarithm conversion for the second sampled system. Minimum realization of normalized transfer function: The normalized transfer function, shown in Eq. (6), is converted to the following SISO controllable minimum realization: x(t) = Ax(t) + bu(t) ry\ y(t) = cx(t) + du(t) Normalized sampled system on sampling points: Integrating the response between two sampling points to the next sampling point, the following precise sampled system is obtained: x((k + \)h) = e kh x(kh) - A" 1 [I - e kh ]bu(kh) y(kh) = cx(kh) + du(kh) Normalized sampled system approximated: Approximating Eq. (3) by the advanced difference method, the following sampled system is obtained: x(k + 1) = (I + Ah)x(k) + bhu(k) y(k) = cx(£) + du(k) Normalized System in continuous region: Returning to the continuous region after conversion using the matrix logarithm function, the following system is obtained in continuous region: x(t) = A*x(0 + b*t/(7) (10) y{t) = cx{t) + du{t) where A* = — ln(I + Ah) h = A-A 2 h + -A 3 h 2 --- (11) b* = (I-A/z + -A 2 /z 2 ---)b Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 265 The condition of convergence for logarithm conversion Eq. (11) of controllable accompany description Eq. (7) is not described because it is assumed that the sampled time h is sufficiently small. The approximated order is then selected as the 9 th order. Thus, d = is assumed for the simplification. 3. Controller and parameter tuning In this section, an IP controller and a number of parameter tuning methods are presented in order to increase the robustness of the control system. 3.1 Normalized IP controller In this section, 3 kinds of description on normalized integral lead dependent type IP controller which is not conventional proportional lead dependent type PI controller are shown. First is showing inherent frequency for normalization as magnitudes of integral and proportional in continuous systems. Second is showing that in digital systems. Third is showing again that of digital systems in returning approximated continuous systems. C(s) = K i (^p) = K i CD n (-^ (12) s s co n c * = K.p(z-l + h/p) = K.p(z-l + ha)* n /p) z -\ z-\ C*(s) = K t (= + p) = ^*(- + 4) (14) s s co n Note that the digital IP controller of Eq. (13) is asymptotic to the proportional control as h approaches zero or p becomes larger. This controller is called IPL tuning. Then, the stable zero = - 1/ p must be placed not in the neighborhood of the system poles for safety. 3.2 Stability of closed loop transfer function In this section, more higher order systems are processed for consideration generally on three tuning region classified by the amplitude of P control parameter using Hurwits approach in example of a second-order system with one-order modelling error. It is guessed that there may be four elementary tuning regions and six combinatorial tuning regions generally in the aspect of Hurwits stability. The following normalized loop transfer function is obtained from the normalized controlled object Eq. (5) and the normalized controller Eq. (12): WCs) = K i (l + W)(js+l)_ (15) es" +2 + p 2r+q s n+1 + - + fts 2 + (K t p + V)s + K, If the original parameters \/i,j,g j > 0,a > are positive, then \/k,fi k > • Assuming p > and K > 0, and that <p(s) 4 e F +2 + /? 2 , + r +1 + • • • + fts 2 + (K t p + Vjs + K, (16) 266 Advances in Reinforcement Learning is a Hurwits polynomial, the stability limits of K can be obtained as a region of p . Then, this region is called a IPL region when p has a maximum lower bound and an IPO region when p =0. The region between zero and the minimum upper bound is called the IPS. The region between the minimum upper bound and the maximum lower bound is called the IPM region. Generally, there are four elementary regions and six combinatorial regions. 3.3 Stationary points investing approach on fraction equation In this section, Stationary Points Investing approach on Fraction Equation for searching local maximum with equality restriction is shown using Lagrange's undecided multiplier approach. Then, multiple same solutions of the independent variable are solved at the stationary points. They can be used to check for mistakes in calculation as self-diagnostics approach. Here, the common normalized control parameters K and p will be obtained in continuous region, which has reduction models reduced from original region. Stationary Points Investing for Traction Equation approach for searching local maximum with equality restriction: \wumH^r ( 17 ) — » solve _ local _ muximum I minimum for co-co s such that |^(y<yj| = l This is the design policy of servo control for wide band width. In particular, \W(o)\ = 1 means that steady state error is 0. Next, Lagrange's undecided multiplier approach is applied to obtain stationary points co with equality restriction using the above u,v notations. Then, the original problem can be converted to the following problem: u(co) v(bj) ► solve local maximum I minimum J + (cd,A) = \W(jcof =^ + A{u(a))-v(w)} (18) where X is a Lagrange multiplier. The necessary conditions for obtaining the local minimum/ maximum of a new function become as folio wings. dJ + (a),A) dco U \0) s )v((O s ) - u(CQ s )V \C0 s ) v(^) 2 + X{u\bJ s )-v\bJ s )} = ( 19 ) dJ\co,X) The following relations are obtained from eqs. (19) and (20): u(w s )-v(w s ) = (20) Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 267 u\co s ) = v\co s )or X- (21) (22) u{w s ) = v{w s ) Solutions of control parameters: Solving these simultaneous equations, the following functions can be obtained: K t J =g(* 9 p 9 &j) (y = 1,2,-, a) where co is the stationary points vector. Multiple solutions of K can be used to check for mistakes in calculation. 3.4 Example of a second-order system with one-order modelling error In this section, an IP control system in continuous design for a second-order original controlled object without one-order sensor and signal conditioner dynamics is assumed for simplicity. The closed loop system with uncertain one-order modeling error is normalized and obtained the stable region of the integral gain in the three tuning region classified by the amplitude of P control parameter using Hurwits approach. Then, the safeness of the only I tuning region and the risk of the large P tuning region are discussed. Moreover, the analytic solutions of stationary points and double same integral gains are obtained using the Stationary Points Investing on Traction Equation approach for the gain curve of a closed loop system. Here, an IP control system for a second-order controlled object without sensor dynamics is assumed. Closed-loop transfer function: K n a>l (23) (24) (25) (26) w(i)=— T ( 27 ) el 4 +(2ge+\)I 3 +(s+2g)s 2 ^^p^^I^K, Stable conditions by Hurwits approach with four parameters: a. In the case of a certain time constant IPL&IPS Common Region: s + ■ 2ga> n s + co 2 n H(s)-- ss + \ 1 -G(s)H(s)-. col K S K (£s + \)(s 2 +2ga) n s + a>; ') _A S >£=0) n £ ®n f,(l + ps)(£l + \) 268 Advances in Reinforcement Learning < K. < max[0, min[fc 2 , k 3 , oo]] (28) r A 2^ 2 + 2^ + l) ^2 = — — (29) [p{4g 2 s + 2^ 2 + 2$- - s) - (2gs + 1) 2 ] (30) + lp{4g l e+2ge l +2g-s}-{2gs+\yr k 3 A l + B^ 2 (, 2 + 2„ + l) (31) where p>0 for 4g 2 s + 2^ 2 + 2^ < ^ IPL, IPS Separate Region: The integral gain stability region is given by Eqs. (28) -(30). o< 2 (2 ^ + 2 1)2 *P (^) 0<P< / , 2 _ (2 f_ + 2 1)2 , _ (PS) (32) 4£- z £ + 2^r + 2^- - £ /or 4^- 2 ^ + 2^ 2 +2^--^>0 It can be proven that k 3 >0 in the IPS region, and k 2 — » oo, k 3 — > oo w/zew /? — > (33) IPO Region: ^2^+2^+1) ^ = Q (2^+l) 2 The IPO region is most safe because it has not zeros, b. In the case of an uncertain positive time constant IPL&IPS Common Region: < K. < max[0, min[& 2 , k 3 ]] when p>0 (35) where k 3 (p, g, s p ) = for 4g 2 s+2gs 2 +2g<s (36) . T 4f(f + l) 7 _ t mm ^ 2 = _ — - when s = 1 (37) P IPL, IPS Separate Region: This region is given by Eq. (32). Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 269 IPO Region: 0<K t <2g(l-g 2 )\ ,— (38) £-=+oo,^>0.707 w/w?« (f„ = ^^>0,0<c<0.707),p = p (l-2^ 2 ) b ^ c. Robust loop gain margin The following loop gain margin is obtained from eqs. (28) through (38) in the cases of certain and uncertain parameters: gm^ (39) where K UL is the upper limit of the stable loop gain K ■ Stable conditions by Hurwits approach with three parameters: The stability conditions will be shown in order to determine the risk of one order modelling error, 0<K Z where p>— (PL) (40) < K { < 2q _ where 0<p<— (P0) (41) l-2q> 2q Hurwits Stability is omitted because h is sufficiently small, although it can be checked using the bilinear transform. Robust loop gain margin: gm = oo (PL _ region) (42) It is risky to increase the loop gain in the IPL region too much, even if the system does not become unstable because a model order error may cause instability in the IPL region. In the IPL region, the sensitivity of the disturbance from the output rises and the flat property of the gain curve is sacrificed, even if the disturbance from the input can be isolated to the output upon increasing the control gain. Frequency transfer function: \WU<5)\ = [- K f 2 {Uw 2 p 2 } (K. - Iqco 1 f +QJ 2 {\-a 2 + K t p} 2 —> solve _ local _ maximum I minmum for co = co s ^ ' such that ^V(jc5 s )\ = \ When the evaluation function is considered to be two variable functions ( a> and k ) an d the stationary point is obtained, the system with the parameters does not satisfy the above stability conditions. 270 Advances in Reinforcement Learning Therefore, only the stationary points in the direction of co will be obtained without considering the evaluation function on K alone. Stationary points and the integral gain: Using the Stationary Points Investing for Fraction Equation approach based on Lagrange's undecided multiplier approach with equality restriction, the following two loop gain equations on x are obtained. Both identities can be used to check for miscalculation. K a =0.5{x 2 +2(2g 2 -\)x + \}l{2g + (x-\)p) K i2 = 0.5{3x 2 + 4(2^ 2 - l)x + 1} l{2g + (2x - \)p) where x = co 2 > (44) (45) Equating the right-hand sides of these equations, the third-order algebraic equation and the solutions for semi-positive stationary points are obtained as follows: x = 0,. = J 2 ^ - 1 X2g-^)_ 1 (46) These points, which are called the first and second stationary points, call the first and second tuning methods, respectively, which specify the points for gain 1. 4. Numerical results In this section, the solutions of double same integral gain for a tuning region at the stationary point of the gain curve of the closed system are shown and checked in some parameter tables on normalized proportional gains and normalized damping coefficients. Moreover, loop gain margins are shown in some parameter tables on uncertain time constants of one-order modeling error and damping coefficients of original controlled objects for some tuning regions contained with safest only I region. 0.9 0.95 1.00 1.05 1.10 0.8 0.8612 1.0496 1.1892 1.3068 1.4116 0.9 0.6999 0.9424 1.0963 1.2197 1.3271 1.0 -99 0.8186 1.0000 1.1335 1.2457 1.1 -99 0.6430 0.8932 1.0446 1.1647 1.2 -99 -99 0.7598 0.3480 1.0812 Table 1. co values for g and p in IPL tuning by the first tuning method 0.9 0.95 1.00 1.05 1.10 0.8 0.7750 1.0063 1.2500 1.5063 1.7750 0.9 0.6889 0.8944 1.1111 1.3389 1.5778 1.0 1.2272 0.8050 1.0000 1.2050 1.4200 1.1 1.1077 0.7318 0.9091 1.0955 1.2909 1.2 1.0149 1.0791 0.8333 1.0042 1.1833 Table 2. K a = K i2 values for g and p in IPL tuning by the first tuning method Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 271 Table 1 lists the stationary points for the first tuning method. Table 2 lists the integration gains (K. l =K i2 ) obtained by substituting Eq. (46) into Eqs. (44) and (45) for various damping coefficients. Table 3 lists the integration gains ( K n = K i2 ) for the second tuning method. 0.9 0.95 1.00 1.05 1.10 1.3 1.0 0.8333 0.7143 0.6250 0.5556 1.4 1.250 1.0 0.8333 0.7143 0.6250 1.5 1.667 1.250 1.0 0.8333 0.7143 1.6 2.50 1.667 1.250 1.0 0.8333 1.7 5.00 2.50 1.667 1.250 1.0 Table 3. K 1 = K i2 values for g and p in IPL tuning by the second tuning method Then, a table of loop gain margins (gm>\) generated by Eq. (39) using the stability limit and the loop gain by the second tuning method on uncertain s in a given region of £ for each controlled ^-by IPL (p =1.5) control is very useful for analysis of robustness. Then, the unstable region, the unstable region, which does not become unstable even if the loop gain becomes larger, and robust stable region in which uncertainty of the time constant, are permitted in the region of s~ . Figure 3 shows a reference step up-down response with unknown input disturbance in the continuous region. The gain for the disturbance step of the IPL tuning is controlled to be approximately 0.38 and the settling time is approximately 6 sec. The robustness on indicial response for the damping coefficient change of ±0.1 is an advantageous property. Considering Zero Order Hold, with an imperfect dead-time compensator using l st -order Pade approximation, the overshoot in the reference step response is larger than that in the original region or that in the continuous region. Ki = 1.0,p=1.5) by Second tuning for Normal 2nd Order Syst From: Referense zita=0.9(-0.1) - - zita=1.0(Nominal) zita=1.1(+0.1) i- !\ O \ I - 1 a v - i o i. 1/ f zita=0.9(-0.1) zita=1.0(Nominal) \ (g = 1± 0. 1, K t = 1. 0,p = 1. 5, w n =\.005,g = l,s = 199.3, k = -0.0050674) Fig. 3. Robustness of IPL tuning for damping coefficient change. Then, Table 4 lists robust loop gain margins (gm > 1 ) using the stability limit by Eq.(37) and the loop gain by the second tuning method on uncertain £ in the region of (0.1 < ~s < 10) for each controlled g (>0.7) by IPL(p =1.5) control. The first gray row shows the area that is also unstable. 272 Advances in Reinforcement Learning Table 5 does the same for each controlled g (>0.4) by IPS(p =0.01). Table 6 does the same for each controlled g (>0.4) by IP0( p =0.0). eps/zita 0.3 0.7 0.8 0.9 1 1.1 1.2 0.1 -2.042 -1.115 1.404 5.124 10.13 16.49 24.28 0.2 -1.412 -0.631 0.788 2.875 5.7 9.33 13.83 1.5 -0.845 -0.28 0.32 1.08 2 3.08 4.32 2.4 -1.019 -0.3 0.326 1.048 1.846 2.702 3.6 3.2 -1.488 -0.325 0.342 1.06 1.8 2.539 3.26 5 -2.128 -0.386 0.383 1.115 1.778 2.357 2.853 10 -4.596 -0.542 0.483 1.26 1.81 2.187 2.448 Table 4. Robust loop gain margins on uncertain s in each region for each controlled g at WL (p-l-5) eps/zita 0.4 0.5 0.6 0.7 0.8 0.1 1.189 1.832 2.599 3.484 4.483 0.6 1.066 1.524 2.021 2.548 3.098 1 1.097 1.492 1.899 2.312 2.729 2.1 1.254 1.556 1.839 2.106 2.362 10 1.717 1.832 1.924 2.003 2.073 Table 5. Robust loop gain margins on uncertain s in each region for each controlled g at IPS (JJ=0.01) eps/zita 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.6857 1.196 1.835 2.594 3.469 4.452 5.538 6.722 0.4p).6556 1.087 1.592 2.156 2.771 3.427 4.118 4.84 0.5 0.6604 1.078 1.556 2.081 2.645 3.24 3.859 4.5 0.6 0.6696 1.075 1.531 2.025 2.547 3.092 3.655 4.231 1 0.7313 1.106 1.5 1.904 2.314 2.727 3.141 3.556 2.1 0.9402 1.264 1.563 1.843 2.109 2.362 2.606 2.843 10 1.5722 1.722 1.835 1.926 2.004 2.073 2.136 2.195 9999 1.9995 2 2 2 2 2 2 2 Table 6. Robust loop gain margins on uncertain £ in each region for each controlled g at IPO (p=0.0) These table data with additional points were converted to the 3D mesh plot as following Fig. 4. As IPO and IPS with very small p are almost equivalent though the equations differ quiet, the number of figures are reduced. It implies validity of both equations. According to the line of worst loop gain margin as the parameter of attenuation in the controlled objects which are described by gray label, this parametric stability margin (PSM) (Bhattacharyya S. P., Chapellat H., and Keel L. H., 1994) is classified to 3 regions in IPS and IPO tuning regions and to 4 regions in IPL tuning regions as shown in Fig.5. We may call the Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 273 larger attenuation region with more than 2 loop gain margin to the strong robust segment region in which region uncertainty time constant of one-order modeling error is allowed in the any region and some change of attenuation is also allowed. (c) p =0.5 (d)^=0.01or0 (a) p=1.5 (b)^=1.0 Fig. 4. Mesh plot of closed loop gain margin Next, we call the larger attenuation region with more than y > 1 and less than 2 loop gain margin to the weak robust segment region in which region uncertainty time constant of one-order modeling error is only allowed in some region over some larger loop gain margin and some larger change of attenuation is not allowed. The third and the forth segment is almost unstable. Especially, notice that the joint of each segment is large bending so that the sensitivity of uncertainty for loop gain margin is larger more than the imagination. (a) p =1.5 (b) p =0.01 (c) p =0 (d) p =1.5, 1.0, 0.5, 0.01 Fig. 5. The various worst lines of loop gain margin in a parameter plane (certain&uncertain) Moreover, the readers had to notice that the strong robust region and weak robust region of IPL is shift to larger damping coefficient region than ones of IPS and IPO. Then, this is also one of risk on IPL tuning region and change of tuning region from IPO or IPS to IPL region. 5. Conclusion In this section, the way to convert this IP control tuning parameters to independent type PI control is presented. Then, parameter tuning policy and the reason adopted the policy on the controller are presented. The good and no good results, limitations and meanings in this chapter are summarized. The closed loop gain curve obtained from the second order example with one-order feedback modeling error implies the butter-worth filter model matching method in higher order systems may be useful. The Hardy space norm with bounded window was defined for I, and robust stability was discussed for MIMO system by an expanssion of small gain theorem under a bounded condition of closed loop systems. 274 Advances in Reinforcement Learning We have obtained first an integral gain leading type of normalized IP controller to facilitate the adjustment results of tuning parameters explaining in the later. The controller is similar that conventional analog controllers are proportional gain type of PI controller. It can be converted easily to independent type of PI controller as used in recent computer controls by adding some converted gains. The policy of the parameter tuning is to make the norm of the closed loop of frequency transfer function contained one-order modeling error with uncertain time constant to become less than 1. The reason of selected the policy is to be able to be similar to the conventional expansion of the small gain theorem and to be possible in PI control. Then, the controller and uncertainty of the model becomes very simple. Moreover, a simple approach for obtaining the solution is proposed by optimization method with equality restriction using Lagrange's undecided multiplier approach for the closed loop frequency transfer function. The stability of the closed loop transfer function was investigated using Hurwits Criteria as the structure of coefficients were known though they contained uncertain time constant. The loop gain margin which was defined as the ratio of the upper stable limit of integral gain and the nominal integral gain, was investigated in the parameter plane of damping coefficient and uncertain time constant. Then, the robust controller is safe in a sense if the robust stable region using the loop gain margin is the single connection and changes continuously in the parameter plane even if the uncertain time constant changes larger in a wide region of damping coefficient and even if the uncertain any adjustment is done. Then, IPO tuning region is most safe and IPL region is most risky. Moreover, it is historically and newly good results that the worst loop gain margin as each damping coefficient approaches to 2 in a larger region of damping coefficients. The worst loop gain margin line in the uncertainty time constant and controlled objects parameters plane had 3 or 4 segments and they were classified strong robust segment region for more than 2 closed loop gain margin and weak robust segment region for more than y > 1 and less than 2 loop gain margin. Moreover, the author was presented also risk of IPL tuning region and the change of tuning region. It was not good results that the analytical solution and the stable region were complicated to obtain for higher order systems with higher order modeling error though they were easy and primary. Then, it was unpractical. 6. Appendix A. Example of a second-order system with lag time and one-order modelling error In this section, for applying the robust PI control concept of this chapter to systems with lag time, the systems with one-order model error are approximated using Fade approximation and only the simple stability region of the integral gain is shown in the special proportional tuning case for simplicity because to obtain the solution of integral gain is difficult. Here, a digital IP control system for a second-order controlled object with lag time L without sensor dynamics is assumed. For simplicity, only special proportional gain case is shown. Transfer functions: Ke- Ls K(l-0.5Ls) Kco 2 n (l-0.5Ls) LriS) = = = — (A.l) Ts + l (Ts + \){Q.5Ls + \) s 2 +2gco n s + CQ 2 n v ' Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 275 co n = . 1 ,^ = 0.5y0.5rZ{(r + 0.5Z)/(0.5rZ)} (A2) vo.srz H(s) = -^- (A3) ss + \ Normalized operation: The normalize operations as same as above mentioned are done as follows. — A S — A T s = — ,L = LfD n to. £ =£CD 1 C(s) = K t (± + p) C{s) =Kp H (U—p) S S ($ n Closed loop transfer function: The closed loop transfer function is obtained using above normalization as follows; _ s (g +2gj+l) W(s) es 4 + (2gs + l)s 3 + (s+2g- 0.5 2 L 2 K t )s 2 + s + K, (A4) 0-0^ (A5) J 2 +2(;5"+l (A6) e.s +1 ^.A^L p = ±p (A8) (A9) s (eJ + \)(s 2 +2;s+1) ( a - 10 ) ^.(l + ^)(l-0.5Z^)(ey + l) ~^ 4 + (2 qE+ l)s 3 + (e+ 2q- 0.5pLK t )J 2 ^ (1+ Z.^- 0.5lZ.)7+ Z. z/ p = 0.5L fen £,.(!- 0.5Ls)(£s+l) (All) */ ^ = z7zen ^ ) = ^(1-0-5^X^ + 1) (A.12) ss 4 + (2gs + 1)5 3 + (J + 2g)s 2 + (1 - 0.5ZX,. )s + K t 276 Advances in Reinforcement Learning Stability analysis by Hurwits Approach 1. p < 0.5L, < K. < min { g+2 f , 1 } , $■ > 0, J > 0.5pL (0.5L-p) {(2gs+\)(2g + s-Q.5 2 l}K i )-s}>K i (2gs+lf when p = 0.5L (A13) 2 ^ 2+2 ^ + 1) 2 ^ >K t when p = 0.5L (A14) (2^ + l){(2^ + l) + 0.5 2 Z 2 } k3 < k2 then < K { < min{^|, 2 g (g 2 +2gg+l) ^^ = Q ^ (A15) 0.5 2 T? {2gs +l){{2gs +1) + Q5 2 J}} In continuous region with one order modelling error, 0<K.< 2g ^ when p = 0.5L (A16) ' (l + o.s 2 ^) Analytical solution of Ki for flat gain curve using Stationary Points Investing for Fraction Equation approach is complicated to obtain, then it is remained for reader's theme. In the future, another approach will be developed for safe and simple robust control. B. Simple soft M/ A station In this section, a configuration of simple soft M/ A station and the feedback control system with the station is shown for a simple safe interlock avoiding dangerous large overshoot. B.l Function and configuration of simple soft M/ A station This appendix describes a simple interlock plan for an simple soft M/A station that has a parameter-identification mode (manual mode) and a control mode (automatic mode). The simple soft M/A station is switched from automatic operation mode to manual operation mode for safety when it is used to switch the identification mode and the control mode and when the value of Pv exceeds the prescribed range. This serves to protect the plant; for example, in the former case, it operates when the integrator of the PID controller varies erratically and the control system malfunctions. In the latter case, it operates when switching from P control with a large steady-state deviation with a high load to PI or PID control, so that the liquid in the tank spillovers. Other dangerous situations are not considered here because they do not fall under general basic control. There have several attempts to arrange and classify the control logic by using a case base. Therefore, the M/A interlock should be enhanced to improve safety and maintainability; this has not yet been achieved for a simple M/A interlock plan (Fig. Al). For safety reasons, automatic operation mode must not be used when changing into manual operation mode by changing the one process value, even if the process value recovers to an appropriate level for automatic operation. Semiautomatic parameter identification and PID control are driven by case-based data for memory of tuners, which have a nest structure for identification. This case-based data memory method can be used for reusing information, and preserving integrity and maintainability for semiautomatic identification and control. The semiautomatic approach is adopted not only to make operation easier but also to enhance safety relative to the fully automatic approach. Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 277 Notation in computer control (Fig. Bl, B3) Pv : Process value Av: Actual value Cv : Control value Mv : Manipulated value Sp: Set point A : Auto M : Manual T : Test Mv at manual mode Conditions OnPv ■Pv' Conditions OnM/A Switch -► Integrated Switching -> Logic Self-holding Logic Switch * Mv at auto mode Mv Fig. Bl A Configuration of Simple Soft M/ A Station B.2 Example of a SISO system Fig. B2 shows the way of using M/ A station in a configuration of a SISO control system. r <*> ,/ Controller c(t) M/A u(t) Controlled *k Object 1 L Sensor SC « ' zft ) y(t) Fig. B2 Configuration of a IP Control System with a M/A Station for a SISO Controlled Object where the transfer function needed in Fig.B2 is as follows. 1. Controlled Object: G(s) -- K Ts + 1 Sensor & Signal Conditioner: G (s)- Controller: C(s) = 05K l2 (- + 0.5L) s Sensor Caribration Gain: 1/ K K c T c s + 1 Normalized Gain before M/A Station: 1 / V05TL Normalized Gain after M/A Station: 1 / K Fig. B3 shows examples of simulated results for 2 kinds of switching mode when Pv becomes higher than a given threshold, (a) shows one to out of service and (b) does to manual mode. In former, Mv is down and Cv is almost hold. In latter, Mv is hold and Cv is down. 278 Advances in Reinforcement Learning Auto^ ^Mv _._--—' ^\^M^f\ WJ*^ Manual Pv *v*^v^/\vva/v Av CV (a) Switching example from auto mode to (b) Switching example from auto mode to out of service by Pv High manual mode by Pv High Fig. B3 Simulation results for 2 kinds of switching mode C. New norm and expansion of small gain theorem In this section, a new range restricted norm of Hardy space with window(Kohonen T. r 1995) H™ is defined for I, of which window is described to notation of norm with superscript w, and a new expansion of small gain theorem based on closed loop system like general H™ control problems and robust sensitivity analysis is shown for applying the robust PI control concept of this chapter to MIMO systems. The robust control was aims soft servo and requested internal stability for a closed loop control system. Then, it was difficult to apply process control systems or hard servo systems which was needed strong robust stability without deviation from the reference value in the steady state like integral terms. The method which sets the maximum value of closed loop gain curve to 1 and the results of this numerical experiments indicated the above sections will imply the following new expansion of small gain theorem which indicates the upper limit of Hardy space norm of a forward element using the upper limit of all uncertain feedback elements for robust stability. For the purpose using unbounded functions in the all real domain on frequency like integral term in the forward element, the domain of Hardy norm of the function concerned on frequency is limited clearly to a section in a positive real one-order space so that the function becomes bounded in the section. Proposition Assuming that feedback transfer function H(s) (with uncertainty) are stable and the following inequality is holds, \\H(s <-,y>l Y (C-l) Moreover , if the negative closed loop system as shown in Fig.C-1 is stable and the following inequality holds, \\W(s G(s) 1 + G(s)H{s) <1 (C-2) Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 279 then the following inequality on the open loop transfer function is hold in a region of frequency. |G(;oo)H(;co£ < -^,y > 1 for aye [oo min ,oo max ] (C-3) In the same feedback system, G(s) holds the following inequality in a region of frequency. N^C^^Y'^ 1 f° r «e[co min ,co max ] (C-4) His) Fig. C-l Configuration of a negative feed back system (proof) Using triangle inequality on separation of norms of summension and inequality on separation of norms of product like Helder's one under a region of frequency [co min ,co max ] , as a domain of the norm of Hardy space with window, the following inequality on the frequency transfer function of G(;co) is obtained from the assumption of the proposition. \\W(j a GO'co) l + G(;co)H(;co) <1 |G(jco£ < ||1 + G(;co)H(;co)|£ < 1 + \\G(j<a)£ |H(;o HN* G(jm) (C-5) (C-6) if |H(;co£'<-<l,y>l then |g(;co)|I (C-7) y- 1 i-INHi: Moreover, the following inequality on open loop frequency transfer function is shown. 1 y- T H|G(7co)|I|H(/co)|:>|G(/co)H(/co)|: (C-; On the inequality of norm, the reverse proposition may be shown though the separation of product of norms in the Hardy space with window are not clear. The sufficient conditions on closed loop stability are not clear. They will remain reader's theme in the future. 280 Advances in Reinforcement Learning D. Parametric robust topics In this section, the following three topics (Bhattacharyya S. P., Chapellat H., and Keel L. H., 1994.) are introduced at first for parametric robust property in static one, dynamic one and stable one as assumptions after linearizing a class of non-linear system to a quasi linear parametric variable (QLPV) model by Taylar expansion using first order reminder term. (M.Katoh, 2010) 1. Continuity for change of parameter Boundary Crossing Theorem 1) fixed order polynomials P(A.,s) 2) continuous polynomials with respect to one parameter A. on a fixed interval I=[a,b]. If P(a,s) has all its roots in S, P(b,s) has at least one root in U, then there exists at least one p in (a,b] such that: a) P(p,s) has all roots in S UdS b) P(p,s) has at least one root in dS P(a,s) P(p,s) Fig. D-l Image of boundary crossing theorem 2. Convex for change of parameter Segment Stable Lemma Let define a segment using two stable polynomials as follows. S,(s)±AS,(s) + (l-A)S 2 (s) [^),^)]^»:^e[0,l]} where S 1 (s), S 2 (s) _ is _ polynomials _of _ deg ree _ n and _ stable _ with _ respect _to _S (D-l) Then, the followings are equivalent: a) The segment [S 1 (s),S 2 (s)] is stable with respect to S %(5>0, for _all _s* e dS; X e [0,1] 3. Worst stability margin for change of parameter Parametric stability margin (PSM) is defined as the worst case stability margin within the parameter variation. It can be applied to a QLPV system of a class of non-linear system. There are non-linear systems such as becoming worse stability margin than linearized system although there are ones with better stability margin than it. There is a case which is characterized by the one parameter m which describes the injection rate of I/O, the interpolation rate of segment or degree of non-linearity. E. Risk and Merit Analysis Let show a summary and enhancing of the risk discussed before sections for safety in the following table. Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error 281 Kinds Evaluation of influence Countermeasure 1) Disconnection of feedback line 2) Overshoot over limit value 1) Spill-over threshold 2) Attack to weak material Auto change to manual mode by M/ A station Auto shut down Change of tuning region from IPS to IPL by making proportional gain to large Grade down of stability region from strong or weak to weak or un-stability Use IPO and not use IPS Not making proportional gain to large in IPS tuning region Change of damping coefficient or inverse of time constant over weak robust limit Grade down of stability region from strong or weak to weak or un-stability Change of tuning region from IPL to IPS or IPO Table E-l Risk analysis for safety It is important to reduce risk as above each one by adequate countermeasures after understanding the property of and the influence for the controlled objects enough. Next, let show a summary and enhancing of the merit and demerit discussed before sections for robust control in the following table, too. Kinds Merit Demerit 1) Steady state error is vanishing as time by effect of integral 1) It is important property in process control and hard servo area It is dislike property in soft servo and robot control because of hardness for disturbance There is a strong robust stability damping region in which the closed loop gain margin for any uncertainty is over 2 and almost not changing. It is uniform safety for some proportional gain tuning region and changing of damping coefficient. For integral loop gain tuning, it recommends the simple limiting sensitivity approach. 1) Because the region is different by proportional gain, there is a risk of grade down by the gain tuning. There is a weak robust stability damping region in which the worst closed loop gain margin for any uncertainty is over given constant. 1) It can specify the grade of robust stability for any uncertainty 1) Because the region is different by proportional gain, there is a risk of grade down by the gain tuning. It is different safety for some proportional gain tuning region. Table E-2 Merit analysis for control It is important to apply to the best application area which the merit can be made and the demerit can be controlled by the wisdom of everyone. 282 Advances in Reinforcement Learning 7. References Bhattacharyya S. P., Chapellat H., and Keel L. H.(1994). Robust Control, The Parametric Approach, Upper Saddle River NJ07458 in USA: Prentice Hall Inc. Katoh M. and Hasegawa H., (1998). Tuning Methods of 2 nd Order Servo by I-PD Control Scheme, Proceedings of The 41st Joint Automatic Control Conference, pp. 111-112. (in Japanese) Katoh M.,(2003). An Integral Design of A Sampled and Continuous Robust Proper Compensator, Proceedings ofCCCT2003, (pdf 000564), Vol. Ill, pp. 226-229. Katoh M.,(2008). Simple Robust Normalized IP Control Design for Unknown Input Disturbance, SICE Annual Conference 2008, August 20-22, The University Electro- Communication, Japan, pp.2871-2876, No. :PR0001/ 08/ 0000-2871 Katoh M., (2009). Loop Gain Margin in Simple Robust Normalized IP Control for Uncertain Parameter of One-Order Model Error, International Journal of Advanced Computer Engineering, Vol.2, No.l, January-June, pp.25-31, ISSN:0974-5785, Serials Publications, New Delhi (India) Katoh M and Imura N., (2009). Double-agent Convoying Scenario Changeable by an Emergent Trigger, Proceedings of the 4 th International Conference on Autonomous Robots and Agents, Feb 10-12, Wellington, New Zealand, pp.442-446 Katoh M. and Fujiwara A., (2010). Simple Robust Stability for PID Control System of an Adjusted System with One-Changeable Parameter and Auto Tuning, International Journal of Advanced Computer Engineering, Vol.3, No.l, ISSN:0974-5785, Serials Publications, New Delhi (India) Katoh M.,(2010). Static and Dynamic Robust Parameters and PI Control Tuning of TV-MITE Model for Controlling the Liquid Level in a Single Tank", TC01-2, SICE Annual Conference 2010, 18/ August TC01-3 Krajewski W., Lepschy A., and Viaro U.,(2004). Designing PI Controllers for Robust Stability and Performance, Institute of Electric and Electronic Engineers Transactions on Control System Technology, Vol. 12, No. 6, pp. 973- 983. Kohonen T.,(1995, 1997). Self-Organizing Maps, Springer Kojori H. A., Lavers J. D., and Dewan S. B.,(1993). A Critical Assessment of the Continuous- System Approximate Methods for the Stability Analysis of a Sampled Data System, Institute of Electric and Electronic Engineers Transactions on Power Electronics, Vol. 8, No. 1, pp. 76-84. Miyamoto S.,(1998). Design of PID Controllers Based on Hoc-Loop Shaping Method and LMI Optimization, Transactions of the Society of Instrument and Control Engineers, Vol. 34, No. 7, pp. 653-659. (in Japanese) Namba R., Yamamoto T., and Kaneda M., (1998). A Design Scheme of Discrete Robust PID Control Systems and Its Application, Transactions on Electrical and Electronic Engineering, Vol. 118-C, No. 3, pp. 320-325. (in Japanese) Olbrot A. W. and Nikodem M.,(1994) . Robust Stabilization: Some Extensions of the Gain Margin Maximization Problem, Institute of Electric and Electronic Engineers Transactions on Automatic Control, Vol. 39, No. 3, pp. 652- 657. Zbou K. with Doyle F. C. and Glover K.,(1996). Robust and Optimal Control, Prentice Hall Inc. Zhau K. and Khargonekar P.P., (1988). An Algebraic Riccati Equation Approach to H Optimization, Systems & Control Letters, 11, pp.85-91. 13 Passive Fault Tolerant Control M. Benosman Mitsubishi Electric Research Laboratories 201 Broadway street, Cambridge, MA 02139* USA 1. Introduction Today, as a result of increasing complexity of industrial automation technologies, fault handling of such automatic systems has become a challenging task. Indeed, although industrial systems are usually designed to perform optimally over time, performance degradation occurs inevitably These are due, for example, to aging of system components, which have to be monitored to prevent system-wide failures. Fault handling is also necessary to allow redesign of the control in such a way to recover, as much as possible, an optimal performance. To this end, researchers in the systems control community have focused on a specific control design strategy, called Fault tolerant control (FTC). Indeed, FTC is aimed at achieving acceptable performance and stability for the safe, i.e. fault-free system as well as for the faulty system. Many methods have been proposed to deal with this problem. For survey papers on FTC, the reader may refer to (5; 38; 53). While the available schemes can be classified into two types, namely passive and active FTC (53), the work presented here falls into the first category of passive FTC. Indeed, active FTC is aimed at ensuring the stability and some performance, possibly degraded, for the post-fault model, and this by reconfiguring on-line the controller, by means of a fault detection and diagnosis (FDD) component that detects, isolates and estimates the current fault (53). Contrary to this active approach, the passive solution consists in using a unique robust controller that, will deal with all the expected faults. The passive FTC approach has the drawback of being reliable only for the class of faults expected and taken into account in the design. However, it has the advantage of avoiding the time delay required in active FTC for on-line fault diagnosis and control reconfiguration (42; 54), which is very important in practical situations where the time window during which the faulty system stay stabilizable is very short, e.g. the unstable double inverted pendulum example (37). In fact, in practical applications, passive FTC complement active FTC schemes. Indeed, passive FTC schemes are necessary during the fault detection and estimation phase (50), to ensure the stability of the faulty system, before switching to active FTC. Several passive FTC methods have been proposed, mainly based on robust theory, e.g. multi-objective linear optimization and LMIs techniques (25), QFT method (47; 48), Hoo (36; 37), absolute stability theory (6), nonlinear regulation theory (10; 11), Lyapunov reconstruction (9) and passivity-based FTC (8). As for active FTC, many methods have been proposed for active linear FTC, e.g. (19; 29; 43; 46; 51; 52), as well as for nonlinear FTC, e.g. (4; 7; 13; 14; 20; 21; 28; 32-35; 39; 41; 49). We consider in this work the problem of fault tolerant control for failures resulting from loss of * E-mail: benosman@merl.com 284 Robust Control, Theory and Applications actuator effectiveness. FTCs dealing with actuator faults are relevant in practical applications and have already been the subject of many publications. For instance, in (43), the case of uncertain linear time-invariant models was studied. The authors treated the problem of actuators stuck at unknown constant values at unknown time instants. The active FTC approach they proposed was based on an output feedback adaptive method. Another active FTC formulation was proposed in (46), where the authors studied the problem of loss of actuator effectiveness in linear discrete-time models. The loss of control effectiveness was estimated via an adaptive Kalman filter. The estimation was complemented by a fault reconfiguration based on the LQG method. In (30), the authors proposed a multiple-controller based FTC for linear uncertain models. They introduced an active FTC scheme that ensured the stability of the system regardless of the decision of FDD. However, as mentioned earlier and as presented for example in (50), the aforementioned active schemes will incur a delay period during which the associate FDD component will have to converge to a best estimate of the fault. During this time period of FDD response delay, it is essential to control the system with a passive fault tolerant controller which is robust against actuator faults so as to ensure at least the stability of the system, before switching to another controller based on the estimated post-fault model, that ensures optimal post-fault performance. In this context, we propose here passive FTC schemes against actuator loss of effectiveness. The results presented here are based on the work of the author introduced in (6; 8). We first consider linear FTC and present some results on passive FTC for loss of effectiveness faults based on absolute stability theory. Next we present an extension of the linear results to some nonlinear models and use passivity theory to write nonlinear fault tolerant controllers. In this chapter several controllers are proposed for different problem settings: a) Linear time invariant (LTI) certain plants, b) uncertain LTI plants, c) LTI models with input saturations, d) nonlinear plants affine in the control with single input, e) general nonlinear models with constant as well as time-varying faults and with input saturation. We underline here that we focus in this chapter on the theoretical developments of the controllers, readers interested in numerical applications should refer to (6; 8). 2. Preliminaries Throughout this chapter we will use the L^ norm denoted ||.||, i.e. for x G R" we define | |x| | = Vx T x. The notation Lfh denotes the standard Lie derivative of a scalar function h(.) along a vector function /(.). Let us introduce now some definitions from (40), that will be frequently used in the sequel. Definition 1 ((40), p.45): The solution x(t,xo) of the system x = f(x), x <G IR n , / locally Lipschitz, is stable conditionally to Z, if xq £ Z and for each e > there exists 5(e) > such that \\xo - xo\\ < 5 and x Q e Z => \\x(t,x Q ) - x(t,x )\\ < e, Vt > 0. If furthermore, there exist t(xq) > 0, s.t. ||x(f,Xo) — x(*/*o)ll =^ 0/ V| |xq — Xrjll < t(xq) and xq G Z, the solution is asymptotically stable conditionally to Z. If v(xq) —> oo, the stability is global. Definition 2 ((40), p.48): Consider the system H : x = f(x,u), y = h(x,u), x e W 1 , u,y G IR m , with zero inputs, i.e. x = f(x, 0), y = h(x, 0) and let Z C IR n be its largest positively invariant set contained in {x £ IR n |y = h(x,0) = 0}. We say that H is globally zero-state detectable (GZSD) if x = is globally asymptotically stable conditionally to Z. If Z = {0}, the system H is zero-state observable (ZSO). Passive Fault Tolerant Control 285 Definition 3 ((40), p. 27): We say that H is dissipative in X C IR H containing x = 0, if there exists a function S(x), S(0) = such that for all x G X S(x) >0andS(x(T))-S(x(0)) < f co(u(t),y(t))dt, for all u G U C IR™ and all T > such that *(*) G X, V t G [0,T]. Where the function co : IR m x R OT — > IR called the supply rate, is locally integrable for every u G IT, i.e. J^ 1 |a;(tt(£),i/(£))|d£ < oo, V *o < fi. S is called the storage function. If the storage function is differentiable the previous conditions writes as S{x{t))<co{u{t),y{t)). The system H is said to be passive if it is dissipative with the supply rate w(u,y) = u T y. Definition 4 ((40), p. 36): We say that H is output feedback passive (OFP(jo)) if it is dissipative with respect to co(u,y) = u T y — py T y for some p G IR. We will also need the following definition to study the case of time-varying faults in Section 8. Definition 5 (24): A function x : [0, oo) — > ]R n is called a limiting solution of the system x = f(t, x), f a smooth vector function, with respect to an unbounded sequence t n in [0, oo), if there exist a compact k C IR n and a sequence {x n : [t n ,oo) — ► k} of solutions of the system such that the associated sequence {x n :—> x n (t + t n )} converges uniformly to x on every compact subset of [0, oo). Also, throughout this paper it is said that a statement P(t) holds almost everywhere (a.e.) if the Lebesgue measure of the set {t G [0, oo) \P(t) is false} is zero. We denote by df the differential of the function / : IR n — > IR. We also mean by semiglobal stability of the equilibrium point x° for the autonomous system x = f(x), x G IR n with / a smooth function, that for each compact set K c IR n containing x°, there exist a locally Lipschitz state feedback, such that x° is asymptotically stable, with a basin of attraction containing K ((44), Definition 3, p. 1445). 3. FTC for known LTI plants First, let us consider linear systems of the form x = Ax + Bocu, (1) where, x G IR n , u G IR m are the state and input vector, respectively, and oc G ]R mxm is a diagonal time variant fault matrix, with diagonal elements ocu(t), i = 1, ..., m s.t., < e\ < otu(t) < 1. The matrices A, B have appropriate dimensions and satisfy the following assumption. Assumption(l): The pair (A,B) is controllable. 3.1 Problem statement Find a state feedback controller u(x) such that the closed-loop controlled system (1) admits x = as a globally uniformly asymptotically (GU A) stable equilibrium point \/oc(t) (s.t. < e\ < ocu(t) < 1). 3.2 Problem solution Hereafter, we will re-write the problem of stabilizing (1), for \/oc(t) s.t., < e\ < ocu(t) < 1, as an absolute stability problem or Lure's problem (2). Let us first recall the definition of sector nonlinearities. 286 Robust Control, Theory and Applications Definition 6 ((22), p. 232): A static function tp : [0,oo) xT^ R m ,s.t. [ip(t,y) -Kiy] T [ip(t,y) - K 2 y] < 0, V(f,y), with X = K 2 - K x = K T > 0, where K x = diag(kl lf ...,kl m ), K 2 = diag{kl\, ...,kl m ), is said to belong to the sector \K\, K 2 \. We can now recall the definition of absolute stability or Lure's problem. Definition 7 (Absolute stability or Lure's problem (22), p. 264): We assume a linear system of the form x = Ax + Bu y = Cx + Du (2) u = -ip(t,y), where, x G IR n , u G IR m , y G IR m , (A,B) controllable, (A,C) observable and ip : [0,oo) x R m —> R m is a static nonlinearity, piecewise continuous in t, locally Lipschitz in y and satisfies a sector condition as defined above. Then, the system (2) is absolutely stable if the origin is GUA stable for any nonlinearity in the given sector. It is absolutely stable within a finite domain if the origin is uniformly asymptotically (UA) stable within a finite domain. We can now introduce the idea used here, which is as follows: Let us associate with the faulty system (1) a virtual output vector y G IR OT (3) (4) (5) x = Ax + Bocu y = Kx, and let us write the controller as an output feedback u= -y. From (3) and (4), we can write the closed-loop system as x = Ax + Bv y = Kx v= -cc(t)y. We have thus transformed the problem of stabilizing (1), for all bounded matrices oc(t), to the problem of stabilizing the system (5) for all oc(t). It is clear that the problem of GUA stabilizing (5) is a Lure's problem in (2), with the linear time varying stationarity tp(t,y) = oc(t)y, and where the 'nonlinearities' admit the sector bounds K\ = &iag(e\, ...,e{), K 2 = I mX m- Based on this formulation we can now solve the problem of passive fault tolerant control of (1) by applying the absolute stability theory (26). We can first write the following result: Proposition 1: Under Assumption 1, the closed-loop of (1) with the static state feedback -Kx, (6) where K is solution of the optimal problem ki: min(Ep*E=a., ; K-ij pA(k) + A t (k)p (c t - ((c T - pfyw- 1 ) 7 -I P>0 K KA PB)W -H <0 rank KA n ~ (7) Passive Fault Tolerant Control 287 for P = P T > 0, W = (D + D T ) 05 and {A(K), B(K), C(K), D(K)} is a minimal realization of the transfer matrix -lj -l (8) G=[I + K{sl - A)~ 1 B] [I + e x x I mxm K(sI - A) admits the origin x = as GUA stable equilibrium point. Proof: We saw that the problem of stabilizing (1) with a static state feedback u = — Kx is equivalent to the stabilization of (5). Studying the stability of (5) is a particular case of Lure's problem defined by (2), with the 'nonlinearity' function ip(t,y) = — oc(t)y associated with the sector bounds K\ = e\ x l mX m r ^2 = txm (introduced in Definition 1). Then based on Theorem 7.1, in ((22), p. 265), we can write that under Assumptionl and the constraint of observability of the pair (A, K), the origin x = is GUA stable equilibrium point for (5), if the matrix transfer function G=[I + G(s)][I + e 1 xI mxm G(s)]-\ where G(s) = K(s I — A)~ 1 B / is strictly positive real (SPR). Now, using the KYP lemma as presented in (Lemma 6.3, (22), p. 240), we can write that a sufficient condition for the GUA stability of x = along the solution of (1) with u = — Kx is the existence of P = P T > 0, L and W, s.t. PA(K) + A T (K)P = -L T L - eP, e > PB(K) = C T (K)-L T W (9) W T W = D(K) + D T (K), where, {A, B, C, D} is a minimal realization of G. Finally, adding to equation (9), the observability condition of the pair (A, K), we arrive at the condition PA(K) + A T (K)P= -L T L PB(K) = C T (K)-L T W W T W = D(K) + D T (K) K KA ■eP, e > rank KA n-\ (10) Next, if we choose W = W T we can write W = (D + D T ) 05 . The second equation in (10) leads to L T = (C T — PB)^ -1 . Finally, from the first equation in (10), we arrive at the following condition on P PA(K) + A T (K)P + (C T - PfyW^dC 7 - PB)^- 1 ) 1 < 0, which is in turn equivalent to the LMI PBW- 1 ' pA(k) + A t (k)p (c t ■ ((C T -PB)W- l ) T -I <0. (11) Thus, to solve equation (10) we can solve the constrained optimal problem 288 Robust Control, Theory and Applications mm(EfcrEp"fc|0 PA(K) + A T (K)P (C^-PBjW" 1 (((^-P^W" 1 ) 7 -I P>0 <0 rank KA KA n-l □ (12) Note that the inequality constraints in (7) can be easily solved by available LMI algorithms, e.g. feasp under Matlab. Furthermore, to solve equation (10), we can propose two other different formulations: 1. Through nonlinear algebraic equations: Choose W = W T which implies by the third equation in (10) that W = (D(K) + D T (X)) - 5 , for any K s.t. PA(K) + A T (K)P- -L l L-eP, e>0, P = P 1 > PB(K) = C T (K)-L T W K KA rank KA n-l (13) To solve (13) we can choose e = e 2 and P = P T P, which leads to the nonlinear algebraic equation F(k ij ,p ij ,l ij ,e)=0, (14) where k(j, i = 1, ...,m, j = 1, ...n, pij, i = 1, ...,n (A e IR nxn ), j = 1, ...n and l^, i = 1, ...,m, j = 1, ...ft are the elements of K, P and L, respectively. Equation (14) can then be resolved by any nonlinear algebraic equations solver, e.g.fsolve under Matlab. 2. Through Algebraic Riccati Equations (ARE): It is well known that the positive real lemma equations, i.e. the first three equations in (10) can be transformed to the following ARE ((3), pp. 270-271): vr D -ifiT DP" 1 dTt *>Tt>-1, P(A-BR- l C) + (A L - C 1 R'^ 1 )P + PBR'^B 1 P + C 1 R~ l C = t (15) where A = A + Q5e.I n xn, R = D(K) + D T (K) > 0. Then, if a solution P = P T > is found for (15) it is also a solution for the first three equation in (10), together with W = -VR in , L= (PB- C T )R~ 1/2 V T , VV T = I. To solve equation (10), we can then solve the constrained optimal problem Passive Fault Tolerant Control 289 min(Efcr^Ii^) P>0 K KA rank (16) _KA n ~ where P is the symmetric solution of the ARE (15), that can be directly computed by available solvers, e.g. care under Matlab. There are other linear controllers for LPV system, that might solve the problem stated in Section 3. 1 e.g. (1). However, the solution proposed here benefits from the simplicity of the formulation based on the absolute stability theory, and allows us to design FTCs for uncertain and saturated LTI plants, as well as nonlinear affine models, as we will see in the sequel. Furthermore, reformulating the FTC problem in the absolute stability theory framework may be applied to solve the FTC problem for several other systems, like infinite dimensional systems, i.e. PDEs models, stochastic systems and systems with delays (see (26) and the references therein). Furthermore, compared to optimal controllers, e.g. LQR, the proposed solution offers greater robustness, since it compensates for the loss of effectiveness over [e\, 1]. Indeed, it is well known that in the time invariant case, optimal controllers like LQR compensates for a loss of effectiveness over [1/2, 1] ((40), pp. 99-102). A larger loss of effectiveness can be covered but at the expense of higher control amplitude ((40), Proposition 3.32, p. 100), which is not desirable in practical situations. Let us consider now the more practical case of LTI plants with parameter uncertainties. 4. FTC for uncertain LTI plants We consider here models with structured uncertainties of the form x = {A + AA)x + (B + AB)<xu, (17) where AAeoA = {AA e K nxn \AA min < AA < AA max , AA min ,AA max e IR nxn }, AB e oB = {AB e 7R nxm \AB min < AB < AB maX/ AB min ,AB max e R" xm }, oc = diag(oc\\, ..., oc mm ) , < e\ < oca < 1 Vz G {1, ...,m}, and A, B, x, u as defined before. 4.1 Problem statement Find a state feedback controller u(x) such that the closed-loop controlled system (17) admits x = as a globally asymptotically (G A) stable equilibrium point \/oc(s.t. < e\ < oca < 1), VAA e oA, AB e oB. 4.2 Problem solution We first re-write the model (17) as follows: x = (A + AA)x + (B + AB)v y = Kx (18) v = —ocy. The formulation given by (18), is an uncertain Lure's problem (as defined in (15) for example). We can write the following result: 290 Robust Control, Theory and Applications Proposition 2: Under Assumption 1, the system (17) admits x = as GA stable equilibrium point, with the static state feedback u = —KH~ 1 x, where K, H are solutions of the LMIs Q + HA T - K T L T B T + AH- BLK < VL e L v , Q = Q T > 0, H > -Q + HAA 7 -K T L T AB T + AAH-ABLK<0, V(AA,AB,L) G oA v x o&° xL v , { ' where, L v is the set containing the vertices of {e\l mxm , I m xm}, and oA v , oB v are the set of vertices of oA, oB respectively. Proof: Under Assumption 1, and using Theorem 5 in ((15), p. 330), we can write the stabilizing static state feedback u = —Kx, where K is such that, for a given H > 0, Q = Q T > we have Q + (A - BLK) T H + H(A - BLK) <0\/LeL v -Q+((AA-ABLK) T H + H(AA-ABLK)) <0V(AA,AB,L) e oA v x oB° x L u , where, L v is the set containing the vertices of {e\l m ^ m , I mX m}, and oA v , oB v are the set of vertices of oA, oB respectively Next, inequalities (20) can be transformed to LMIs by defining the new variables K = KH -1 , H = H _1 , Q = H~ 1 QH~ 1 and multiplying both sides of the inequalities in (20) by H _1 , we can write finally (20) as Q + HA T - K T L T B T + AH- BLK < VL e L v , Q = Q T > 0, H > -Q + HAA T - K T L T AB T + AAH - ABLK < V(AA, AB,L) G oA v x o&° xL v , { ' the controller gain will be given by K = XH _1 .D Let us consider now the practical problem of input saturation. Indeed, in practical applications the available actuators have limited maximum amplitudes. For this reason, it is more realistic to consider bounded control amplitudes in the design of the fault tolerant controller. 5. FTC for LTI plants with control saturation We consider here the system (1) with input constraints \iij\ < u maXi , i = 1, •••, m, and study the following FTC problem. 5.1 Problem statement Find a bounded feedback controller, i.e. \u{\ < u maXi , i = 1, •••, m, such that the closed-loop controlled system (1) admits x = as a uniformly asymptotically (UA) stable equilibrium point \/oc(t) (s.t. < e\ < ocii(t) < 1), i = 1, ...,m, within an estimated domain of attraction. 5.2 Problem solution Under the actuator constraint \u{\ < u maXi , i = 1, ...,m, the system (1) can be re-written as x = Ax + BU max v y = Kx (22) v = —oc(t)sat(y), where U max = diag(u maXl/ ...,u m ax m ), sat(y) = (saf(y 1 ),...,sflf(y m )) T , sat(yi) = sign(yi)min{l,\yi\}. Thus we have rewritten the system (1) as a MIMO Lure's problem with a generalized sector condition, which is a generalization of the SISO case presented in (16). Next, we define the two functions tpi : IR n — > IR m , tf\(x) = —e\l mxm sat(Kx) and Passive Fault Tolerant Control 291 xp 2 : IR n -> IR m , t^ 2 (x) = -sat(Kx). We can then write that v is spanned by the two functions tpi, xp2'. v(x,t) e co{xp 1 (x),xp 2 (x)}, \/x eR n ,t e R, (23) where co{ipi(x), ^i{x)} denotes the convex hull of xpi, tp2, i.e. i=2 i=2 co{xp 1 (x) / xp 2 (x)} := {£7i(0*iW/ E^'W = l > Ti(0 > W}. i=l z=l Note that in the SISO case, the problem of analyzing the stability of x = for the system (22) under the constraint (23) is a Lure's problem with a generalized sector condition as defined in (16). Let us recall now some material from (16; 17), that we will use to prove Proposition 4. Definition 8 ((16), p.538): The ellipsoid level set e(P,p) := {x e IR n : V(x) = x T Px < p}, p > 0, P = P T > is said to be contractive invariant for (22) if V = 2x T P(Ax - BU max ocsat(Kx)) < 0, forallxG£(P,p)\{0}, VfeR. Proposition 3 ((26), P. 539): An ellipsoid e(P,p) is contractively invariant for x = Ax + Bsat(Fx), B G R nxl if and only if (A + BF) T P + P(A + BF) < 0, and there exists an H £ R lxn such that (A + BH) T P + P(A + BH) < 0, and e(P,p) c{xeK N : \Fx\ < 1}. Fact 1 ((16), p. 539): Given a level set Ly(p) = {x e IR n / V(x) < p} and a set of functions ipi(u), i e {1,...,N}. Suppose that for each i e {1, ..., N}, Ly(p) is contractively invariant for x = Ax + Btpi(u). Let tp(u,t) G co{\pi(u), i e {1,...,N}} for all w,f G IR, then Ly(p) is contractively invariant for x = Ax + Bxp(u, t). Theorem 1((17), p. 353): Given an ellipsoid level set z(P,p), if there exists a matrix H G K mxn such that (A + BM(v,K,H)) T P + P(A + BM(v,K,H)) < 0, for all 1 e G V := {z? G IR n |z? z - = 1 or 0}, and e(P,p) C £(H) := {x G R N : |/z z -x| < 1, i = 1, ...,m}, where ^l + (l-t?i)fei M(r; / jK / H)= : _v m k m + (1 — v m )h m then s(P,p) is a contractive domain for x = Ax + Bs«f(Xx). We can now write the following result: Proposition 4: Under Assumption 1, the system (1) admits x = as a UA stable equilibrium (24) 1 Hereafter, hi, kj denote the zth line of H, K, respectively. 292 Robust Control, Theory and Applications point, within the estimated domain of attraction e(P,p), with the static state feedback u Kx = YQ~ 1 x / where Y, Q solve the LMI problem > 0, / > «*/q>o,y,gJ 7* I' I Q QA T + AQ + M(v, Y, G) T (BU n QA T + AQ + M(v,Y,G) T (BU n 1 gi g Q > 0, i oc e ) T + (BU max ct e )M(v,Y,G) < 0, Vz? e V (25) ) T + (BUmax)M(v,Y,G) < 0, \/v e V e\ x Imxm, M given by (24), P = pQ ^ and where g/ G IR lxn is the z'th line of G, a: 6 K > 0, p > are chosen. Proof: Based on Theorem 1 recalled above, the following inequalities (A + BU max cteM(v, -K,H)) T P + P(A + BU max cceM(v, -K,H)) < 0, (26) together with the condition e(P,p) C £(H) are sufficient to ensure that e(P,p) is contractive invariant for (1) with oc = e\l m ^ m , u = —u max sat(Kx). Again based on Theorem 1, the following inequalities (A + BU max M(v, -K,H)) T P + P(A + BU max M{v, -K,H)) < 0, (27) together with e(P,p) C C(H) are sufficient to ensure that s(P,p) is contractive invariant for (1) with oc = Imxm, u = —u max sat(Kx). Now based on the direct extension to the MIMO case, of Fact 1 recalled above, we conclude that e(P,p) is contractive invariant for (1) with U = —U m a X Sat(Kx), WoCa(t), i = 1, ...,171, S.t.,0 < €\ < &u(t) < 1. Next, the inequalities conditions (26), (27) under the constraint e(P,p) C C(H) can be transformed to LMI conditions ((17), p. 355) as follows: To find the control gain K such that we have the bigger estimation of the attraction domain, we can solve the LMI problem in fQ>0,Y,GJ I Q > 0, / > lax ct e ) T + (BU max ct e )M(v,Y,G) <0,\/veV XJ T + (BU max )M(v, Y, G) < 0, \/v e V (28) QA T + AQ + M(v, Y, G) T {BU t QA T + AQ + M(v, Y, G) T (BU-, \ g l >0, i = l,...,m, si Q\ ~ where Y = -KQ, Q = (P/p)" 1 , G = H^P/p)' 1 , M(v,Y,G) = M(v,-K,H)Q, g { = hiiP/p)- 1 , hi e R lxn is the z'th line of H and R > is chosen. □ Remark 1: To solve the problem (25) we have to deal with 2 m+1 + m + 1 LMIs, to reduce the number of LMIs we can force Y = G, which means K = —H(P/p)~ 1 Q~ 1 . Indeed, in this case the second and third conditions in (25) reduce to the two LMIs QA T + AQ + G T (BU n + (BU m axCLe)G<0 QA T + AQ + G T (BU max ) T + (BU max )G < 0, (29) which reduces the total number of LMIs in (25) to m + 3. ♦ In the next section, we report some results in the extension of the previous linear controllers to single input nonlinear affine plants. Passive Fault Tolerant Control 293 6. FTC for nonlinear single input affine plants Let us consider now the nonlinear affine system x = f(x)+g(x)ocu, (30) where x G IR n , u G IR represent, respectively, the state vector and the scalar input. The vector fields /, columns of g are supposed to satisfy the classical smoothness assumptions, with /(0) = 0. The fault coefficient is such that < e\ < oc < 1. 6.1 Problem statement Find a state feedback controller u(x) such that the closed-loop controlled system (44) admits x = as a local (global) asymptotically stable equilibrium point Va: (s.t. < e\ < oc < 1). 6.2 Problem solution We follow here the same idea used above for the linear case, and associate with the faulty system (44) a virtual scalar output, the corresponding system writes as (31) x = f(x) + g(x)ocu y = k(x), where k : IR — > IR is a continuous function. Let us chose now the controller as the simple output feedback u = -k(x). (32) We can then write from (31) and (32) the closed-loop system as x = f(x) + g(x)v y = k{x) (33) v = —ocy. As before we have cast the problem of stabilizing (44), for all oc as an absolute stability problem (33) as defined in ((40), p. 55). We can then use the absolute stability theory to solve the problem. Proposition 5: The closed-loop system (44) with the static state feedback u = -k(x), (34) where k is such that there exist a C 1 function S : K n — > IR positive semidefinite, radially unbounded, i.e. S(x) —> +oo, \\x\\ —> +oo, that satisfies the PDEs L f S(x) = -0.5q T (x)q(x) + (j^) k 2 (> L g S(x)=(]±^)k(x)- q T w, (35) where the function w : IR n — > R z is s.t. w T w = t^t, and a : R n —> R z , I G N, under the condition of local (global) detectability of the system x = f(x)+g(x)v y = k{x), (36) 294 Robust Control, Theory and Applications admits the origin x = as a local (global) asymptotically stable equilibrium point. Proof: We saw the equivalence between the problem of stabilizing (44), and the absolute stability problem (33), with the 'nonlinearities' sector bounds e\ and 1. Based on this, we can use the sufficient condition provided in Proposition 2.38 in ((40), p. 55) to ensure the absolute stability of the origin x = of (33), for all a: G [e\, 1]. First we have to ensure that the parallel interconnection of the system x = f{x)+g(x)v y = *(*), (3/) with the trivial unitary gain system y = v, (38) is OFP(— k), with k = y^r- and with a C 1 radially unbounded storage function S. Based on Definition 4, this is true if the parallel interconnection of (37) and (38) is dissipative with respect to the supply rate w{v,y) = v T y + [yzTf) fv> (39) where y = y + v. This means, based on Definition 3, that it exists a C 1 function S : IR n — > IR, with S(0) = and S(x) > 0, Vx, s.t. S(x(t)) < co(v,y) ,2 (40) <^y+||p|| 2 +( T ^)||y + p| Furthermore, S should be radially unbounded. From the condition (40) and Theorem 2.39 in ((40), p. 56), we can write the following condition on S, k for the dissipativity of the parallel interconnection of (37) and (38) with respect to the supply rate (39): L f S(x) = -Q.5q T (x)q(x) + (^l) k 2 (x) L g S(x)=k(x)+2^)k(x)-q T w, where the function w : IR n — > M. 1 is s.t. w T w = y^r, and q : IR" — > R z , I G N. Finally, based on Proposition 2.38 in ((40), p. 55), to ensure the local (global) asymptotic stability of x = 0, the system (37) has to be locally (globally) ZSD, which is imposed by the local (global) detectability of (36). □ Solving the condition (41) might be computationally demanding, since it requires to solve a system of PDEs. We can simplify the static state feedback controller, by considering a lower bound of the condition (40). Indeed, condition (40) is true if the inequality S < v T y, (42) is satisfied. Thus, it suffices to ensure that the system (37) is passive with the storage function S. Now, based again on the necessary and sufficient condition given in Theorem 2.39 ((40), p. 56), the storage function and the feedback gain have to satisfy the condition LfS(x)<0 L g S(x) = k(x). K ' Passive Fault Tolerant Control 295 V = u £ = (XV m = o £ , * = /(*)+£(*)£ X u(0) = Fig. 1. The model (45) in cascade form However, by considering a lower bound of (40), we are considering the extreme case where e\ —> 0, which may result in a conservative feedback gain (refer to (6)). It is worth noting that in that case the controller given by (52), (41), reduces to the classical damping or Jurdjevic-Quinn control u = —L g S(x), e.g. ((40), p. Ill), but based on a semidefinite function S. 7. FTC for nonlinear multi-input affine plants with constant loss of effectiveness actuator faults We consider here affine nonlinear models of the form ■f(x)+g(x)u, (44) where x G IR n , u G IR m represent respectively the state and the input vectors. The vector fields /, and the columns of g are assumed to be C 1 , with /(0) = 0. We study actuator's faults modelled by a multiplicative constant coefficient, i.e. a loss of effectiveness, which implies the following form for the faulty model 2 ■f(x)+g(x)ccu, (45) 1, ..., m s.t., where a G ]R mxm is a diagonal constant matrix, with the diagonal elements &#, i < e\ < oca < 1. We write then the FTC problem as follows. Problem statement: Find a feedback controller such that the closed-loop controlled system (45) admits x = as a globally asymptotically stable (GAS) equilibrium point \/cc (s.t. < e\ < oca < 1). 7.1 Problem solution Let us first rewrite the faulty model (45) in the following cascade form (see figure 1) x = f(x)+g(x)h(Z) £ = uv, £(0) = y = HO = Z, (46) where we define the virtual input v = u with w(0) = 0. This is indeed, a cascade form where the controlling subsystem, i.e. £ dynamics, is linear (40). Using this cascade form, it is possible to write a stabilizing controller for the faulty model (45), as follows. 2 Hereafter, we will denote by x the states of the faulty system (45) to avoid cumbersome notations. However, we remind the reader that the solutions of the healthy system (44) and the faulty system (45) are different. 296 Robust Control, Theory and Applications Theorem 2: Consider the closed-loop system that consists of the faulty system (45) and the dynamic state feedback u = -L g W(x) T -k$,u(0)=0 t = ei (-(L g W(x)) T -kZ),ao)=0, ( ' where W is a C 1 radially unbounded, positive semidefinite function, s.t. LfW < 0, and k > 0. Consider the fictitious system * = /(*)+£(*)£ ^ = e 1 (-(L g W) T + v) (48) y = m = e. If the system (48) is (G)ZSD with the input v and the output y, then the closed-loop system (45) with (47) admits the origin (x,£) = (0,0) as (G)AS equilibrium point. Proof: We first prove that the cascade system (48) is passive from v to y = £. To do so, let us first consider the linear part of the cascade system t = e x v, f (0) = y = *»(£) = I { ] The system (49) is passive, with the C 1 positive definite, radially unbounded, storage function IT(£) = ^£ T £- Indeed, we can easily see that V T > U (S(T)) = \?(T)e(T) < J q v T ydt <1 [ T $ T £dt €\ JO 1 fZ( T ) T 11 T < - / ^< ^-^(T)f(T), which is true for < e\ < 1 Next, we can verify that the nonlinear part of the cascade x = f(x)+g(x)S y = L g W(x), is passive, with the C 1 radially unbounded, positive semidefinite storage function W. Since, W = LfW + i-^W^ < L^W^. Thus we have proved that both the linear and the nonlinear parts of the cascade are passive , we can then conclude that the feedback interconnection (48) of (49) and (50) (see figure 2) is passive from the new input v = L g W + v to the output £, with the storage function S(x, £) = W(x) + U(f ) (see Theorem 2.10 in (40), p. 33). Finally, the passivity associated with the (G)ZSD implies that the control v = — kg, k > achieves (G)AS (Theorem 2.28 in (40), p. 49). Up to now, we proved that the negative feedback output feedback v = — k£, k > achieves the desired AS for oc = e\l m ^ m . We have to prove now that the result holds for all oc s.t. < e\ < oca < 1, even if ^ is fed back from the fault's model (46) with oc = e\l m ^ m , since we do not know the actual value of a. If we multiply the control law (47) by a constant gain matrix k = diag(k\, ...,k m ), 1 < k\ < ^-, we can write the new control as u = -~k(L g W(x) T -kZ),k>0 l «(0) = . . <t = e 1 k(-(L g W(x)) T -kt),m=0. ( ' Passive Fault Tolerant Control 297 V f I X — f(x) -\- Q~(x)^ V = L g W(x) Fig. 2. Feedback interconnection of (49) and (50) It is easy to see that this gain does not change the stability result, since we can define for the nonlinear cascade part (50) the new storage function W = kW and the passivity is still satisfied from its input £ to the new output kLgW(x). Next, since the ZSD property remains unchanged, we can chose the new stabilizing output feedback v = —kk^, which is still a stabilizing feedback with the new gain kk > 0, and thus the stability result holds for all oc s.t. < e\ < oca < 1, i = 1, ...,m. □ The stability result obtained in Theorem 2, depends on the ZSD property Indeed, if the ZSD is global, the stability obtained is global otherwise only local stability is ensured. Furthermore, we note here that with the dynamic controller (47) we ensure that the initial control is zero, regardless of the initial value of the states. This might be important for practical applications, where an abrupt switch from zero to a non zero initial value of the control is not tolerated by the actuators. In Theorem 2, one of the necessary conditions is the existence of W > 0, s.t. the uncontrolled part of (45) satisfies LfW < 0. To avoid this condition that may not be satisfied for some practical systems, we propose the following Theorem. Theorem 3: Consider the closed-loop system that consists of the faulty system (45) and the dynamic state feedback -^W T + i Sf (f + g£)), p = diag(j5 n ,...,l3 n *< -*(£-/HC(*)) .), < | < ft,- < 1, *(f - ftC(x)) - pL g W> + ftf (/ + g), f (0) = 0, «(0) = 0, (52) where k > and the C 1 function K(x) is s.t. there exists a C 1 radially unbounded, positive semidefinite function W satisfying dW -fo(f(x)+g(x)pK(x)) < 0, Vx e R H , V/3 = diag(p n ,...,Pmm), < e x < fe < 1. (53) Consider the fictitious system x=f(x)+g(x)£ 9 = s-m*)- (54) If (54) is (G)ZSD with the input 9 and the output y, for for all p s.t. ft,-, i = 1, ..., m, < e\ < Pa < 1- Then, the closed-loop system (45) with (52) admits the origin (x, £) = (0, 0) as (G)AS 298 Robust Control, Theory and Applications equilibrium point. Proof: We will first prove that the controller (52) achieves the stability results for a faulty model with oc = e\l m ^ m and then we will prove that the stability result holds the same for all oc s.t. oca, i = 1, ...,m, < e\ < oca < 1. First, let us define the virtual output y = £ — f$K(x), we can write the model (46) with oc = x = f(x)+g(x)(y + pK(x)) t = e\hnKmV (55) y = Z-j5K(x), we can then write dK ij = erfmxmv - £— (/ + g(y + pK(x))) = v. To study the passivity of (55), we define the positive semidefinite storage function 1 r,Tr, and write V = pW(x) + -fy, V = fSL f+gfiK W + pL g Wy + fv, and using the condition (53), we can write V<f(pL g W T + v), which establishes the passivity of (55) from the new input v = v + ftLgW T to the output y. Finally, using the (G)ZSD condition for oc = e\I m xm, we conclude about the (G)AS of (46) for oc = €iI mX m, with the controller (52) (Theorem 2.28 in (40), p. 49). Now, remains to prove that the same result holds for all oc s.t. oca, i = 1, ..., m, < e\ < oca < 1, i.e. the controller (52) has the appropriate gain margin. In our particular case, it is straightforward to analyse the gain margin of (52), since if we multiply the controller in (52) by a matrix oc, s.t. oca, i = 1, ..., m, < e\ < oca < 1, the new control writes as u = ii mx »(-«fc(f - m*)) - «p l sW t + «/*i (/ + so), k > 0, ,B = diag(Pi,...,p m ), < ft < 1, (56) t = <xm-pK{x))-apL g W T + a^ (/ + g?), f(0) = 0, M (0) = 0. We can see that this factor will not change the structure of the initial control (52), since it will be directly absorbed by the gains, i.e. we can write k = ock, with all the elements of diagonal matrix k positive, we can also define ft = ocfi which is still a diagonal matrix with bounded elements in [ei,l], s.t. (53) and (54) are is still satisfied. Thus the stability result remains unchanged. □ The previous theorems may guaranty global AS. However, the conditions required may be difficult to satisfy for some systems. We present below a control law ensuring, under less demanding conditions, semiglobal stability instead of global stability. Theorem 4: Consider the closed-loop system that consists of the faulty system (45) and the dynamic state feedback U = -fc(£ - U n0 m(x)), k > 0, t = -ke 1 (Z-u nom (x)),m=Q, «(0)=0 / { ] Passive Fault Tolerant Control 299 where the nominal controller u nom (x) achieves semiglobal asymptotic and local exponential stability of x = for the safe system (44). Then, the closed-loop (45) with (57) admits the origin (x, £) = (0, 0) as semiglobal AS equilibrium point. Proof: The prove is a direct application of the Proposition 6.5 in ((40), p. 244), to the system (46), with oc = eilmxm- Any positive gain oc, s.t. 1 < a# < ^-, i = 1, ...,m, will be absorbed by k > 0, keeping the stability results unchanged. Thus the control law (57) stabilize (46) and equivalently (45) for all oc s.t. oca, i = 1, ...,m, < e\ < oca < 1. D Let us consider now the practical problem of input saturation. Indeed, in practical systems the actuator powers are limited, and thus the control amplitude bounds should be taken into account in the controller design. To solve this problem, we consider a more general model than the affine model (44). In the following we first study the problem of FTC with input saturation, on the general model x = f(x)+g(x,u)u, (58) where, x, u, f are defined as before, g is now function of both the states and the inputs, and is assumed to be C 1 w.t.r. to x, u. The actuator faut model, writes as x = f(x) + g(x,ocu)ccu, (59) with the loss of effectiveness matrix oc defined as before. This problem is treated in the following Theorem, for the scalar case where oc € \e\, 1], i.e. when the same fault occurs on all the actuators. Theorem 5: Consider the closed-loop system that consists of the faulty system (59), for oc G \e\, 1], and the static state feedback u(x) = -A(x)G(x,0) T G(*,0) = ^ lg (*,0) AW = (l+7i(|xP+4I7 2 |G(x%P))(l+|G(x / 0)P) > ° 71 - Jo 1+^(1) aS 7i(s) = H 2s (-riW-i)^ + s where W is a C 2 radially unbounded, positive semidefinite function, s.t. LfW < 0. Consider the fictitious system x = f(x)+g(x,e 1 u)e 1 u dW(x) ( N ( 61 ) If (61) is (G)ZSD, then the closed-loop system (59) with (60) admits the origin as (G)AS equilibrium point. Furthermore \u(x)\ <u, Vx. Proof: Let us first consider the faulty model (59) with oc = e\ . For this model, we can compute the derivative of W as dW(x) W(x) = L f W+ v ) - e x g(x,e x u)u dW(x) W(x) < — ^^eig(x,eiu)u. Now, using Lemma II.4 (p. 1562 in (31)), we can directly write the controller (60), s.t. W < -1a(x)|G(x,0)| 2 . 300 Robust Control, Theory and Applications Furthermore \u(x)\ <u, \/x. We conclude then that the trajectories of the closed-loop equations converge to the invariant set{x| A(x)|G(x,0)| 2 = 0} which is equivalent to the set {x\ G(x,0) = 0}. Based on Theorem 2.21 (p. 43, (40)), and the assumption of (G)ZSD for (61), we conclude about the (G)AS of the origin of (59), (60), with oc = e\. Now multiplying u by any positive coefficient oc, s.t. < e\ < a < 1 does not change the stability result. Furthermore, if \u(x)\ < u, Vx, then \au(x) | < u, Vx, which completes the proof. □ Remark 2: In Theorem 5, we consider only the case of scalar fault oc £ [e\, 1], i.e. the case of uniform fault, since we need this assumption to be able to apply the result of Lemma II.4 in (31). However, this assumption can be satisfied in practice by a class of actuators, namely pneumatically driven diaphragm-type actuators (23), for which the failure of the pressure supply system might lead to a uniform fault of all the actuators. Furthermore, in Proposition 6 below we treat for the case of systems affine in the control, i.e. g(x, u) = g(x), the general case of any diagonal matrix of loss of effectiveness coefficients. Proposition 6: Consider the closed-loop system that consists of the faulty system (45), and the static state feedback u(x) = -A(x)G(x) T G(x) = ^W« (62) A ( x ) = i+|Goor where W is a C 2 radially unbounded, positive semidefinite function, s.t. LfW < 0. Consider the fictitious system x = f{x)+g(x)e 1 u dW(x) f x I 63 ) If (63) is (G)ZSD, then the closed-loop system (45) with (62) admits the origin as (G)AS equilibrium point. Furthermore \u(x)\ <u, \/x. Proof: The proof follows the same steps as in the proof of Theorem 5, except that in this case the constraint of considering that the same fault occurs on all the actuators, i.e. for a scalar oc, is relaxed. Indeed, in this case we can directly ensure the negativeness of W, since if u is such that W < — A(x)L g W (x)eiLgW (x) T < 0, then in the case of a diagonal fault matrix, the derivative writes as W < -A{x)L g W(x)e 1 aL g W(x) T < -X(x)e\L g W(x)L g ]N(x) T < -e\\(x)\G(x)\ 2 . Thus, the stability result remains unchanged. □ Up to now we have considered the case of abrupt faults, modelled with constant loss of effectiveness matrices. However, in practical applications, the faults are usually time-varying or incipient, modelled with time-varying loss of effectiveness coefficients, e.g. (50). We consider in the following section this case of time-varying loss of effectiveness matrices. 8. FTC for nonlinear multi-input affine plants with time-varying loss of effectiveness actuator faults We consider here faulty models of the form x = f(x)+g(x)oc(t)u, (64) where oc(t) is a diagonal time-varying matrix, with C 1 diagonal elements ocjj(t), i = \,...,m s.t., < e\ < ocuit) < 1, \/t. We write then the FTC problem as follows. Problem statement: Find a feedback controller such that the closed-loop controlled system (64) admits x = as a uniformly asymptotically stable (UAS) equilibrium point \/cx(t) (s.t. < e\ < ocu(t) < 1). Passive Fault Tolerant Control 301 8.1 Problem solution To solve this problem we use some of the tools introduced in (24), where a generalization of Krasovskii-LaSalle theorem, has been proposed for nonlinear time-varying systems. We can first write the following result. Theorem 6: Consider the closed-loop system that consists of the faulty system (64) with the dynamic state feedback u = -L g W(x) T -k^ f k> 0, m(0) = ff = a(0(-(L g w(x)) T -kf) / f(o) = o / m where oc(t) is a C 1 function, s.t. < e\ < oc(t) < 1, W, and W is a C 1 , positive semidefinite function, such that: 1- L f W < 0, 2- The system x = f(x) is AS conditionally to the set M = {x \ W(x) = 0}, 3- V(x, J) limiting solutions for the system * = /(*)+*(*)£ £ = «(f)(-(L g W) T -^) (66) y = *(*,£)=£, w.r.t. unbounded sequence {f n } in [0, oo), then if h{x, £) = 0, a.e., then either (x, £) (£q) = (0, 0) for some to > or (0, 0) is a a;-limit point of (x, £), i.e. Hindoo (*, £) (0 — > (0, 0). Then the closed-loop system (64) with (65) admits the origin (x, £) = (0, 0) as UAS equilibrium point. Proof: Let us first rewrite the system (64) for oc(t) = oc{t), in the cascade form * = /(*) + *(*)*(£) f = &(t)v, v = ic, £(0) = 0, m(0) = (67) 3/ = ft(f ) = f- Replacing v = uby its value in (65) gives the feedback system * = /(*) +*(*)/«(£) j = «(f)(-L ? W(x) T + 0), J(0) = 0, m(0) = (68) V = *(? ) = f- We prove that (68) is passive from the input v to the output £. We consider first the linear part of (67) ff = aW r ff (0) = y = fc(£) = & which is passive with the storage function IT(£) = 2? T £/ i- e - ^(^/£) — £ T £ — £ T &(t) v < Next, we consider the nonlinear part * = /(*)+#(*)£ , 7m y = W*), (/U) which is passive with the storage function W(x), s.t. W = LfW + L^W£ < LgWf . We conclude that the feedback interconnection (68) of (69) and (70) is passive from v to f , with 302 Robust Control, Theory and Applications the storage function S(x, f ) = W(x) + 11(f) (see Theorem 2.10, p. 33 in (40)). This implies that the derivative of S along (68) with v = — k£, k > 0, writes s(t,x,€)<e T e<o. Now we define for (68) with v = — kg, k > 0, the positive invariant set M ={(*,£) | W(z) + lT(£) = 0} M= {(x,0)|W(x) = 0}. We note that the restriction of (68) with v = — fcf, k > on M is x = f(x), then applying Theorem 5 in (18), we conclude that, under Condition 2 of Theorem 6, the origin (x, f ) = (0, 0) is US for the system (64) for oc = oc and the dynamic controller (65). Now, multiplying u by any oc(t), s.t. < e\ < &u(t) < 1, \/t, does not change neither the passivity property, nor the AS condition of x = f(x) on M, which implies the US of (x, J) = (0, 0) for (64), (65) \/oc(t), s.t. < € 1 < ocu{t) < 1, W. Now we first note the following fact: for any a > and any t > t q we can write S(t,x(t),at))-S(t ,x(t ),at0)) < - [' Jl(h(S(T)))dT = - [' k\S(T)\ 2 dT, J to J to thus we have f\ H (h(ar))) - a)dx < f ti(h($(T)))dT < S(t ,x(t ),Z(t )) < M;M > 0. J to J t Finally, using Theorem 1 in (24), under Condition 3 of Theorem 6, we conclude that (x,£) = (0,0)isUASfor(64),(65). □ Remark 3: The function cc in (65) has been chosen to be any C 1 time varying function, s.t. < e\ < cc(t) < 1, \/t. The general time-varying nature of the function was necessary in the proof to be able to use the results of Theorem 5 in (18) to prove the US of the faulty system's equilibrium point. However, in practice one can simply chose oc(t) = 1, \/t ♦. Remark 4: Condition 3 in Theorem 6 is general and has been used to properly prove the stability results in the time-varying case. However, in practical application it can be further simplified, using the notion of reduced limiting system. Indeed, using Theorem 3 and Lemma 7 in (24), Condition 3 simplifies to: V(x, f ) solutions for the reduced limiting system i = /(x)+g(*)f £ = ^(t)(-(L g W(x)) T -*£) (71) where the limiting function oc<y(t) is defined us oc<y(t) = lim n ^oo oc(t + t n ) w.r.t. unbounded sequence {t n } in [0,oo). Then, if h(x,^) = 0, a.e., then either (x,£)(£q) = (0/0) for some to > or (0, 0) is a a;-limit point of (x, £ ). Now, since in our case the diagonal matrix-valued function oc is s.t. < e\ < ocu(t) < 1, \/t, then it obviously satisfies a permanent excitation (PE) condition of the form / t+T (x(t)oc(t) 1 dr > rl, T > 0, r > 0, W, t Passive Fault Tolerant Control 303 which implies, based on Lemma 8 in (24), that to check Condition 3 we only need to check the classical ZSD condition: Vx solutions for the system x = f{x) L g W(x) = 0, K/L) either x(t§) = for some £q > or is a o;-limit point of x. ♦ Let us consider again the problem of input saturation. We consider here again the more general model (58), and study the problem of FTC with input saturation for the time-varying faulty model x = f(x) + g{x,cc{t)u)cc{t)u, (73) with the diagonal loss of effectiveness matrix oc(t) defined as before. This problem is treated in the following Theorem, for the scalar case where oc(t) 6 [e\, 1], W, i.e. when the same fault occurs on all the actuators. Theorem 7: Consider the closed-loop system that consists of the faulty system (73) for oc € [e\, 1], W, with the static state feedback u(x) = -A(x)G(x,0) T G(x,0) = ^(x,0) AW = (l+ 7 i(|x| 2 +4U 2 |G(x 1 !o)|2))(l+|G(x / 0)|2) > ° 71 -Jo 1+^(1)^ ji(s) = u: s (ii(t)-i)dt+ s ~ / \ fi,r^ dW(x) d%(x,Teiu) j -^ where W is a C 2 , positive semidefinite function, such that: 1- L f W < 0, 2- The system x = f(x) is AS conditionally to the set M = {x \ W(x) = 0}, 3- Vx limiting solutions for the system x = f{x) +g ( x ,e 1 u(x))(-HxHt)^(xMx,0)) T ' ' I dW i y = h(x)=A(x) - 5 \™( X )g(x,0)\, (75) w.r.t. unbounded sequence {t n } in [0, oo), then if h(x) = 0, a.e., then either x(^o) — f° r some to > or is a o;-limit point of x. Then the closed-loop system (73) with (74) admits the origin x = as UAS equilibrium point. Furthermore \u(x)\ <u, Vx. Proof: We first can write, based on Condition 1 in Theorem 7 dW W< -^—g(x,oc(t)u)oc(t)u, using Lemma II.4 in (31), and considering the controller (74), we have W < -yA(x)|G(x,0)| 2 , \u{x)\ <u\/x. Next, we define for (73) and the controller (74) the positive invariant setM= {x\ W(x) = 0}. Note that we can also write M = {x\ W(x) =0}^ {x\ G(x,0) = 0} <& {x\ u(x) = 0}. 304 Robust Control, Theory and Applications Thus, the restriction of (73) on M is the system x = f(x). Finally, using Theorem 5 in (18), and under Condition 2 in Theorem 7, we conclude that x = is US for (73) and the controller (74). Furthermore if \u(x)\ < u then \cc(t)u(x)\ < u^t r x. Now we note that for the virtual output y = h(x) = A(x) - 5 1 ^ (x)g(x, 0) |, and a > we can write W(t,x(t))-W(t ,x(t ))<-% f \y(r)\ 2 dr = - f\{y(r))dr f 2 Jt J t thus we have /'(p(y(T)) - a)dr < f F (y(r))dr < W(t ,x(t )) J t J t < M, M > 0. Finally, based on this last inequality and under Condition 3 in Theorem 7, using Theorem 1 in (24), we conclude that x = is UAS equilibrium point for (73), (74). □ Remark 5: Here again we can simplify Condition 3 of Theorem 7, as follows. Based on Proposition 3 and Lemma 7 in (24), this condition is equivalent to: Vx solutions for the reduced limiting system j = f(x) +g(x,e l u(x))(-Mxy 7 (t)^r(x)g(x,0)) T (7M y = h(x)=A(x) - 5 \™(x)g(x,0)\, ( ' where the limiting function oc<y(t) is defined us a.j(t) = lim n ^oo oc(t + t n ) w.r.t. unbounded sequence {t n } in [0, oo). Then, if h(x) = 0, a.e., then either x(to) = for some t q > or is a a; -limit point of x. Which writes directly as the ZSD condition: Vx solutions for the system * = f ^ (77) either x(to) = for some to > or is a 6t;-limit point of x. ♦ Theorem 7 deals with the case of the general nonlinear model (73). For the particular case of affine nonlinear models, i.e. g(x, u) = g(x), we can directly write the following Proposition. Proposition 7: Consider the closed-loop system that consists of the faulty system (64) with the static state feedback u(x) = -\(x)G(x) T G(x) = ^(x) (78) A *^ X ) = l+|G(x)| 2 - where W is a C 2 , positive semidefinite function, such that: 1- L f W < 0, 2- The system x = f(x) is AS conditionally to the setM= {x \ W(x) = 0}, 3- Vx limiting solutions for the system x = f{x)+g(x){-\{xy{t)™(x)g(x)y y = h(x)=A(x) - 5 \^(x)g(x)\, ( ' w.r.t. unbounded sequence {t n } in [0, oo), then if h(x) = 0, a.e., then either x(£q) = for some to > or is a a;-limit point of x. Then the closed-loop system (64) with (78) admits the origin x = as UAS equilibrium point. Furthermore \u(x)\ <u, Vx. Proof: The proof is a direct consequence of Theorem 7. However in this case the constraint of Passive Fault Tolerant Control 305 considering that the same fault occurs on all the actuators, is relaxed. Indeed, in this case we can directly write Va(f) G K mxm , s.t. < e 1 < oc u (t) < 1, W: W < -A(x)L g W(x)oc(t)L g W(x) T W < -A(x)e 1 L g W(x)L g W(x) T < -e^G^ 1 . The rest of the proof remains unchanged. □ If we compare the dynamic controllers proposed in the Theorems 2, 3, 4, 6 and the static controllers of Theorems 5, 7, we can see that the dynamic controllers ensure that the control at the initialization time is zero, whereas this is not true for the static controllers. In the opposite, the static controllers have the advantage to ensure that the feedback control amplitude stays within the desired bound u. We can also notice that, except for the controller in Theorem 3, all the remaining controllers proposed here do not involve the vector field / in there computation. This implies that these controllers are robust with respect to any uncertainty Af as long as the conditions on /, required in the different theorems are still satisfied by the uncertain vector field / + Af. Furthermore, the dynamic controller of Theorem 4 inherits the same robustness properties of the nominal controller u nom used to write equation (57) (refer to Proposition 6.5, (40), p. 244). 9. Conclusion and future work In this chapter we have presented different passive fault tolerant controllers for linear as well as for nonlinear models. Firstly, we have formulated the FTC problem in the context of the absolute stability theory, which has led to direct solutions to the passive FTC problem for LTI systems with uncertainties as well as input saturations. Open problems to which this formulation may be applied include infinite dimension models, stochastic models as well as time-delay models. 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Issues on integration of fault diagnosis and reconfigurable control in active fault-tolerant control systems. In 6th IFAC Symposium on fault detection supervision and safety of technical processes, pages 1513-1524, China, August 2006. 14 Design Principles of Active Robust Fault Tolerant Control Systems Anna Filasova and Dusan Krokavec Technical University ofKosice Slovakia 1. Introduction The complexity of control systems requires the fault tolerance schemes to provide control of the faulty system. The fault tolerant systems are that one of the more fruitful applications with potential significance for those domains in which control must proceed while the controlled system is operative and testing opportunities are limited by given operational considerations. The real problem is usually to fix the system with faults so that it can continue its mission for some time with some limitations of functionality. These large problems are known as the fault detection, identification and reconfiguration (FDIR) systems. The practical benefits of the integrated approach to FDIR seem to be considerable, especially when knowledge of the available fault isolations and the system reconfigurations is used to reduce the cost and to increase the control reliability and utility. Reconfiguration can be viewed as the task to select these elements whose reconfiguration is sufficient to do the acceptable behavior of the system. If an FDIR system is designed properly, it will be able to deal with the specified faults and maintain the system stability and acceptable level of performance in the presence of faults. The essential aspect for the design of fault-tolerant control requires the conception of diagnosis procedures that can solve the fault detection and isolation problem. The fault detection is understood as a problem of making a binary decision either that something has gone wrong or that everything is in order. The procedure composes residual signal generation (signals that contain information about the failures or defects) followed by their evaluation within decision functions, and it is usually achieved designing a system which, by processing input /output data, is able generating the residual signals, detect the presence of an incipient fault and isolate it. In principle, in order to achieve fault tolerance, some redundancy is necessary. So far direct redundancy is realized by redundancy in multiple hardware channels, fault-tolerant control involve functional redundancy. Functional (analytical) redundancy is usually achieved by design of such subsystems, which functionality is derived from system model and can be realized using algorithmic (software) redundancy. Thus, analytical redundancy most often means the use of functional relations between system variables and residuals are derived from implicit information in functional or analytical relationships, which exist between measurements taken from the process, and a process model. In this sense a residual is a fault indicator, based on a deviation between measurements and model-equation-based computation and model based diagnosis use models to obtain residual signals that are as a rule zero in the fault free case and non-zero otherwise. 31 Robust Control, Theory and Applications A fault in the fault diagnosis systems can be detected and isolated when has to cause a residual change and subsequent analyze of residuals have to provide information about faulty component localization. From this point of view the fault decision information is capable in a suitable format to specify possible control structure class to facilitate the appropriate adaptation of the control feedback laws. Whereas diagnosis is the problem of identifying elements whose abnormality is sufficient to explain an observed malfunction, reconfiguration can be viewed as a problem of identifying elements whose in a new structure are sufficient to restore acceptable behavior of the system. 1.1 Fault tolerant control Main task to be tackled in achieving fault-tolerance is design a controller with suitable reconfigurable structure to guarantee stability, satisfactory performance and plant operation economy in nominal operational conditions, but also in some components malfunction. Generally, fault-tolerant control is a strategy for reliable and highly efficient control law design, and includes fault-tolerant system requirements analysis, analytical redundancy design (fault isolation principles) and fault accommodation design (fault control requirements and reconfigurable control strategy). The benefits result from this characterization give a unified framework that should facilitate the development of an integrated theory of FDIR and control (fault- tolerant control systems (FTCS)) to design systems having the ability to accommodate component failures automatically. FTCS can be classified into two types: passive and active. In passive FTCS, fix controllers are used and designed in such way to be robust against a class of presumed faults. To ensure this a closed-loop system remains insensitive to certain faults using constant controller parameters and without use of on-line fault information. Because a passive FTCS has to maintain the system stability under various component failures, from the performance viewpoint, the designed controller has to be very conservative. From typical relationships between the optimality and the robustness, it is very difficult for a passive FTCS to be optimal from the performance point of view alone. Active FTCS react to the system component failures actively by reconfiguring control actions so that the stability and acceptable (possibly partially degraded, graceful) performance of the entire system can be maintained. To achieve a successful control system reconfiguration, this approach relies heavily on a real-time fault detection scheme for the most up-to-date information about the status of the system and the operating conditions of its components. To reschedule controller function a fixed structure is modified to account for uncontrollable changes in the system and unanticipated faults. Even though, an active FTCS has the potential to produce less conservative performance. The critical issue facing any active FTCS is that there is only a limited amount of reaction time available to perform fault detection and control system reconfiguration. Given the fact of limited amount of time and information, it is highly desirable to design a FTCS that possesses the guaranteed stability property as in a passive FTCS, but also with the performance optimization attribute as in an active FTCS. Selected useful publications, especially interesting books on this topic (Blanke et al.,2003), (Chen and Patton,1999), (Chiang et al.,2001), (Ding,2008), (Ducard,2009), (Simani et al.,2003) are presented in References. Design Principles of Active Robust Fault Tolerant Control Systems 31 1 1.2 Motivation A number of problems that arise in state control can be reduced to a handful of standard convex and quasi-convex problems that involve matrix inequalities. It is known that the optimal solution can be computed by using interior point methods (Nesterov and Nemirovsky,1994) which converge in polynomial time with respect to the problem size and efficient interior point algorithms have recently been developed for and further development of algorithms for these standard problems is an area of active research. For this approach, the stability conditions may be expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of powerful numerical solvers. Some progres review in this field can be found e.g. in (Boyd et al.,1994), (Herrmann et al.,2007), (Skelton et al.,1998), and the references therein. In contradiction to the standard pole placement methods application in active FTCS design there don't exist so much structures to solve this problem using LMI approach (e.g. see (Chen et al.,1999), (Filasova and Krokavec,2009), (Liao et al.,2002), (Noura et al.,2009)). To generalize properties of non-expansive systems formulated as Hoo problems in the bounded real lemma (BRL) form, the main motivation of this chapter is to present reformulated design method for virtual sensor control design in FTCS structures, as well as the state estimator based active control structures for single actuator faults in the continuous-time linear MIMO systems. To start work with this formalism structure residual generators are designed at first to demonstrate the application suitability of the unified algebraic approach in these design tasks. LMI based design conditions are outlined generally to posse the sufficient conditions for a solution. The used structure is motivated by the standard ones (Dong et al.,2009), and in this presented form enables to design systems with the reconfigurable controller structures. 2. Problem description Through this chapter the task is concerned with the computation of reconfigurable feedback u(£), which control the observable and controllable faulty linear dynamic system given by the set of equations q(0=Aq(0 + B„u(0 + B / f(0 (1) y(0 = Cq(0 + D M u(0 + D / f(0 (2) where q(t) e R n , n(t) e R r ', y(t) e R m , and i(t) G R l are vectors of the state, input, output and fault variables, respectively, matrices A e R nxn , B u e R nxr , C € j^mxn^ D w e R mxr ,Bj: e R nxl ,Dj: e R mxl are real matrices. Problem of the interest is to design the asymptotically stable closed-loop systems with the linear memoryless state feedback controllers of the form u(f) = -K y e (f) (3) u(f) = -Kq c (f)-Lf c (f) (4) respectively. Here K 6 R rxm is the output controller gain matrix, K 6 R rxn is the nominal state controller gain matrix, L 6 R rxl is the compensate controller gain matrix, y e (r) is by virtual sensor estimated output of the system, q e (f) € R" is the system state estimate vector, and ( e (t) e R 1 is the fault estimate vector. Active compensate method can be applied for such systems, where D / D„ (5) 312 Robust Control, Theory and Applications and the additive term B ci (t) is compensated by the term -B f i e (t) = -B u U e (t) (6) which implies (4). The estimators are then given by the set of the state equations q e (f) = Aq e (r) + B„u(f) + B f f e (t) + J(y(f) - y e (f)) (7) f e (f) = Mf c (f)+N(y(f)-y c (f)) (8) y e (f) = Cq e (f)+D H u(f) + D / f e (r) (9) where J 6 R" xm is the state estimator gain matrix, and M € R lxl , N € j^ xm are the system and input matrices of the fault estimator, respectively or by the set of equation q /c (f) = Aq fe (t) + B u u f (t) + J( y/ (f) - D„ U/ (f) - C/q /e (f)) (10) y e (t) = E(y / (f) + (C-EC / )q /e (f) (11) where E £ R mxm is a switching matrix, generally used in such a way that E = 0, or E = I m . 3. Basic preliminaries Definition 1 (Null space) Let E, E e K /zx/z / rank(E) = k < hbe a rank deficient matrix. Then the null space A/e o/E fs fe orthogonal complement of the row space ofE. Proposition 1 (Orthogonal complement) Let E, E e R hxh , rank(E) = k < h be a rank deficient matrix. Then an orthogonal complement E- 1 ofE is E°U4 (12) where Uj is the null space ofE and E° is an arbitrary matrix of appropriate dimension. Proof. The singular value decomposition (SVD) of E, E £ R hxh , rank(E) = k < h gives U T EV E [Vx V 2 ] El o 12 "21 "22 (13) where U T G R hxh is the orthogonal matrix of the left singular vectors, V 6 R hxh is the orthogonal matrix of the right singular vectors of E and L^ £ R kxk is the diagonal positive definite matrix of the form :diag[(7i ■•• cr*], a"! > ••• >a k >0 (14) which diagonal elements are the singular values of E. Using orthogonal properties of U and V, i.e. U T U = l h , as well as V T V = l h , and u, r [Ui u 2 ] II o I 2 ujui respectively, where 1^ G R E = ULV T hxh is the identity matrix, then E can be written as [Ui u 2 ] Si 12 21 22 vj [Ui u 2 ] UiS 1^1 (15) (16) Design Principles of Active Robust Fault Tolerant Control Systems 313 where Si = EiV-[ . Thus, (15) and (16) implies uJe = Uj [Ui u 2 ] It is evident that for an arbitrary matrix E° is = (17) (18) (19) respectively, which implies (12). This concludes the proof. ■ Proposition 2. (Schur Complement) Let Q > 0, R > 0, S are real matrices of appropriate dimensions, then the next inequalities are equivalent E°ujE = E X E = E ± = E°U? Q S S T -R < <& Proof. Let the linear matrix inequality takes form Q + SR _1 S r -R IcT Q s S T -R then using Gauss elimination principle it yields <0 <& Q + SR _1 S J <0, R >0 <0 I SR _1 I R Since det I R X S T I I SR r I Q + SR^S 1 -R (20) (21) (22) (23) and it is evident that this transform doesn't change negativity of (21), and so (22) implies (20). This concludes the proof. ■ Note that in the next the matrix notations E, Q, R, S, U, and V be used in another context, too. Proposition 3 (Bounded real lemma) For given 7 € R and the linear system (1), (2) with i(t) = if there exists symmetric positive definite matrix P > such that A T P + PA PB W C T <0 (24) where l r G K rxr , \ m e R mxm are the identity matrices, respectively then given system is asymptotically stable. Hereafter, * denotes the symmetric item in a symmetric matrix. Proof. Defining Lyapunov function as follows v(q(t)) = q T (*)Pq(0 + f* (y T (r)y(r) - 7 2 u T (r)u(r))dr > (25) where P = P T > 0, P £ R nxn ,y e R, and evaluating the derivative ofv(q(t)) with respect to t then it yields v(q(t)) = q T (f)Pq(0 + q T (t)Pq(t) + y T (t)y(t) - 7 2 u T (f)u(f) < (26) 314 Robust Control, Theory and Applications Thus, substituting (1), (2) with i(t) = it can be written v(q(t)) = (Aq(0 + B w u(f)) T Pq(f) + q T (t)T> (Aq(t) + B w u(f)) + + (Cq(f) + D M u(f)) T (Cq(f) + D M u(*)) - J 2 u T (t)u(t) < and with notation it is obtained where Since P c = qJ(0=[q T «" r W] *(q(0) = q c T (f)P c q c (f) < o A T P + PA PB„ * -7 2 I r _ + C T C C T D„ * D^D U <0 C T C C T D W Schur complement property implies D [CD,] >0 C T >0 (27) (28) (29) (30) (31) (32) then using (32) the LMI (30) can now be written compactly as (24). This concludes the proof. I Remark 1 (Lyapunov inequality) Considering Lyapunov function of the form ^(q«) = q T (0Pq« > o where P = P T > 0, P € R nxn , and the control law u(f) = -K (y(f) - D u u(r)) = -K Cq(r) where K 6 R rxm is a gain matrix. Because in this case (27) gives v(q(t)) = (Aq(f) + B„u(f)) T Pq(f) + q T (f)P(Aq(f) + B„u(f)) < then inserting (34) into (35) it can be obtained *(q(0) = i T (t)Pcbi(t) < o where (33) (34) (35) (36) (37) P cb = A 1 P + PA - PB U K C - (PB„K C) ' < Especially, if all system state variables are measurable the control policy can be defined as follows u(f) = -Kq(f) (38) and (37) can be written as A T P + PA - PB U K - (PB U K) T < Note that in a real physical dynamic plant model usually D u = 0. (39) Design Principles of Active Robust Fault Tolerant Control Systems 315 Proposition 4 Let for given real matrices F, G and O = O t > of appropriate dimension a matrix A has to satisfy the inequality FAG T + GA T F T - < (40) then any solution of A can be generated using a solution of inequality * T FH + GA T1 -H FHF i <0 (41) where H = H T > is a free design parameter. Proof. If (40) yields then there exists a matrix H _1 = H~ T > such that FAG T + GA T F T - + GA T H X AG T < Completing the square in (42) it can be obtained (FH + GA T )H" 1 (FH + GA T ) T ■ and using Schur complement (43) implies (41). FHF <0 (42) (43) 4. Fault isolation 4.1 Structured residual generators of sensor faults 4.1.1 Set of the state estimators To design structured residual generators of sensor faults based on the state estimators, all actuators are assumed to be fault-free and each estimator is driven by all system inputs and all but one system outputs. In that sense it is possible according with given nominal fault-free system model (1), (2) to define the set of structured estimators for k = 1, 2, ... ,m as follows qjteto = Afe,qfe,(0 + B M]ke u(0 + JsJkT s jk(y(0 - D M u(0) (44) y ke (t) = Cq ke (t) + D u u(t) (45) where A ke e R nxn , B uke e R nxr , J sk e R nx ( m ~ 1 ) / and T sk e Rijn-l)xm takes the next form "1 ••• 00000 ••• 00" l sk '■m0k ' 00 00 1000 000 10 00 ••• 00000 00 00 1 (46) Note that T s £ can be obtained by deleting the k-th row in identity matrix I m . Since the state estimate error is defined as e^(f) = q(t) — c^ e {t) then ejt(t) = Aq(f) + B„u(f) - A fe q fe (f) - B ufe u(f) - ] sk T sk {y{t) - D„u(f)) = (A - A ke - J sfc T sfc C)q(f) + (B„ - B ufe )u(f) + A^e^t) To obtain the state estimate error autonomous it can be set Afe = A - J s jtT sfc C, B uke = B„ (47) (48) 31 6 Robust Control, Theory and Applications It is obvious that (48) implies h(t) = A ke e k (t) = (A - ] sk T sk C)e k (t) (49) (44) can be rewritten as qjte(0 = ( A " h^skCHke(t) + B„u(t) + J sfc T sfc (y(t) - D„u(r)) = = Aqfc(t) + B„u(r) + J sk T sk (y(t) - (Cq fe (f) + D u u(f))) and (44), (45) can be rewritten equivalently as qfc(') = Aq fe (f) + B„u(f) + J sfc T sfc (y(f) - y ke (t)) (51) YfeC) = Cqfa(t) + D„u(f) (52) Theorem 1 The k-th state-space estimator (52), (53) is stable if there exist a positive definite symmetric matrix P sk > 0, P sk € R nxn and a matrix Z sk e R nx ( m ~ l ) such that ? sk = P s \ > (53) ^ T Vsk + PsfcA Then ] sk can be computed as A T P sit + P sfc A - Z sit T sfc C - C T T&Z£ < (54) J sfc = P^Zrt (55) Proof. Since the estimate error is autonomous Lyapunov function of the form v(e k (t)) = el(t)F sk e k (t) > (56) where F sk = Pj fc > 0, F sk e R nxn can be considered. Thus, v(e k (t)) = el(t) (A - hkTskC) Tl P sk e k (t) + el(t)F 8k (A - h^ s kC)e k (t) < (57) v(e k (t)) = el(t)F skc e k (t)<0 (58) respectively, where Psfcc = A T P sfc + P sfc A - F sk ] sk T sk C - (F sk ] sk T sk C) T < (59) Using notation F sk ] sk = Z sk (59) implies (54). This concludes the proof. ■ 4.1.2 Set of the residual generators Exploiting the model-based properties of state estimators the set of residual generators can be considered as r s *(0 = X sjk q te (0 + Y sJfc (y(0 - D M u(*)), * = 1,2, . . ., m (60) Subsequently r s *(0 = X sjk (q(0 - ejk(0) + V sfc Cq(f) = (X sfc + Y sJk C)q(0 - X sk e k (t) (61) Design Principles of Active Robust Fault Tolerant Control Systems 317 -y t (t) - y.(t) y,(t) y 2 (t) - j I : v_ Fig. 1. Measurable outputs for single sensor faults To eliminate influences of the state variable vector it is necessary in (61) to consider X sfc + Y sfc C = (62) Choosing X sk = — T sfc C (62) implies *sfc = -T sfc C / Y sk = T sk (63) Thus, the set of residuals (60) takes the form 's*(0 = T sjk (y(0 - D M u(0 - Cq ke (t)), * = 1,2, . . . , m (64) When all actuators are fault-free and a fault occurs in the Z-th sensor the residuals will satisfy the isolation logic \\*sk(t)\\<h sk ,k = l, \\i s k(t)\\>Kk,k^l (65) This residual set can only isolate a single sensor fault at the same time. The principle can be generalized based on a regrouping of faults in such way that each residual will be designed to be sensitive to one group of sensor faults and insensitive to others. Illustrative example To demonstrate algorithm properties it was assumed that the system is given by (1), (2) where the nominal system parameters are given as 1 1 3 1 , Vu = 2 1 5 -9 -5 1 5 1 2 1 1 1 D tt 00 00 and it is obvious that T 8 i = l20l= [Ol],T s2 = I 20 2= [10] T sl C= [110], T s2 C= [12 1" Solving (53), (54) with respect to the LMI matrix variables V sk , and Z sk using Self-Dual-Minimization (SeDuMi) package for Matlab, the estimator gain matrix design problem was feasible with the results L'sl 0.8258 -0.0656 0.0032 -0.0656 0.8541 0.0563 0.0032 0.0563 0.2199 0.6343 0.2242 -0.8595 0.8312 0.5950 -4.0738 318 Robust Control, Theory and Applications Fig. 2. Residuals for the 1st sensor fault / — r 2 (t)- / : 1 : | : " r,(t) — r 2 (t) u -5 -10 -15 -20 -25 -30 -35 15 20 25 ig. 3. Residuals for the 2nd sensor fault 0.8258 -0.0656 0.0032" 0.0335 " 0.1412 Ps2 = -0.0656 0.8541 0.0563 / z s2 = 0.6344 , hi = 1.0479 0.0032 0.0563 0.2199 -0.9214 -4.4614 respectively. It is easily verified that the system matrices of state estimators are stable with the eigenvalue spectra p(A-J s iT sl C) = {-1.0000 -2.3459 -3.0804} p(A-J s2 T s2 C) = {-1.5130 -1.0000 -0.2626} respectively, and the set of residuals takes the form r s i(0=[0 1] y(0 *s2(0=[io] y(0 1 2 1 1 1 1 2 1 1 1 qjte(') qjte(0 Fig. 1-3 plot the residuals variable trajectories over the duration of the system run. The results show that one residual profile remain about the same through the entire run while the second shows step changes, which can be used in the fault isolation stage. Design Principles of Active Robust Fault Tolerant Control Systems 31 9 4.2 Structured residual generators of actuator faults 4.2.1 Set of the state estimators To design structured residual generators of actuator faults based on the state estimators, all sensors are assumed to be fault-free and each estimator is driven by all system outputs and all but one system inputs. To obtain this a congruence transform matrix T ak £ R nxn , k = 1,2,... , r be introduced, and so it is natural to write To*q(0 = T ak Aq(t) + T ak B u n(t) (66) q fc (0 = A Jk q(0 + B Mjk u(0 (67) respectively, where as well as T flfc A / B wfc = T flfc B w (68) y k (t) = CT ak q(t) = Cq k (t) (69) The set of state estimators associated with (67), (69) for k = 1, 2, . . . , r can be defined in the next form q ke (t) = A k q ke (t) + B uke u(t) + l*y(0 - ] k y ke (t) (70) 7ke(t) = Cqfa(0 (71) A ke £ R" x " / B Hfce € R" xr / J t/ L fc € K" xm . Denoting the estimate error as e k (t) = q k (t)-q ke (t) the next differential equations can be written h(t) = qjt(0 - qfe( f ) = = Ajtq(t) + B uJfc u(f) - AjtqjteCf) - B Hfce u(f) - Ljty(f) + J fc y fe (f) = = A iq (t) + B Hfc u(f) - A k (q k (t) - e k (t)) - B uke u(t)- (72) -L l Cq(t)+JfcC(qjfc(t)-e fc (f)) = = (A k - A k T ak + ] k CT ak - L k C)q(t) + (B uk - B uke )u(t) + (A k - J k C)e k (t) h(t) = (T afc A - A ke T ak - LfcC)q(t) + (B uk - B uke )u(t) + A ke e k (t) (73) respectively, where A fe = A i -J i C = T flt A-J fc C, fc = l,2,...,r (74) are elements of the set of estimators system matrices. It is evident, to make estimate error autonomous that it have to be satisfied LfcC = T flfc A - A ke T ak , B uke = B uk = T afc B w (75) Using (75) the equation (73) can be rewritten as e fc (0 = A ke e k (t) = (Ajt - J k C)e k (t) = (T ak A - ] k C)e k (t) (76) and the state equation of estimators are then qjte(0 = ( T akA - J k C)q ke (t) + B uk u(t) + L k y(t) - ] k y ke (t) (77) Yke(t) = Cqfc(t) (78) 320 Robust Control, Theory and Applications 4.2.2 Congruence transform matrices Generally, the fault-free system equations (1), (2) can be rewritten as q(*) = Aq(*) + b MJk u jk (*)+ E h uh*h(t) (79) h=l,h^k r y(f) = Cq(f) + D H u(f) = CAq(f) + Cb H)t u / t(f)+D H u(f) + £ Cb uh u h (t) (80) h=l,hjtk Cb uk u k (t)=y{t)-CAq{t)-D u u(t)- £ Ch «hMt) (81) respectively. Thus, using matrix pseudoinverse it yields u k (t) = (Cb uk ) el (y(t)-CAq(t)-D u u(t)- £ Cb uft u ft (f)) (82) h=l,h^k and substituting (81) b Bl u fc (t) = b Hfc (Cb ui ) el Cb Hfc u t (t) (83) (I n - b uk (Cb uk ) el C)b uk u k (t) = (84) respectively. It is evident that if Tak = I» - b Mfc (Cb wfc ) el C , fc = 1,2, . . . ,r (85) influence of u^(f) in (77) be suppressed (the k-th column in B uk = T afc B w is the null column, approximatively ) . 4.2.3 Estimator stability Theorem 2 The k-th state-space estimator (77), (78) is stable if there exist a positive definite symmetric matrix V ak > 0, V ak e R nxn and a matrix Z ak e R nxm such that F ak = F T ak > (86) A T T^P afc + I Then ] k can be computed as A T T flfc P afc + P, fc T flfc A - Z flfc C - C T Z fl T fc < (87) J k = V~^Z ak (88) Proof. Since the estimate error is autonomous Lyapunov function of the form v(e k (t)) = el(t)P ak e k (t) > (89) where F ak = P T ak > 0, F ak eR™ can be considered. Thus, J7(ejt(0) = e[(t)(T fljt A - } k C) T V ak e k {t) + e[(f)P flfc (T fljt A - ] k C)e k (t) < (90) v(e k (t)) = eJ(t)P akc e k (t) < (91) respectively, where F akc = A T T T ak P ak + P a)t T fl/t A - P flt J t C - (P ak hC) T < (92) Using notation P ak J k = Z ak (92) implies (87). This concludes the proof. ■ Design Principles of Active Robust Fault Tolerant Control Systems 321 4.2.4 Estimator gain matrices Knowing ] k , k = 1, 2, ... , r elements of this set can be inserted into (75). Thus L fc C = A* - A ke T ak = A* - (A* - J fc C) (I - b ujt (Cb„ fc ) el C) = = (h+(M-hC)Kk(Cb uk ) el )c=(h + *keKk(Cb uk ) el )c and (93) L k = h + A ke b uk (Cb uk )^, fc = l,2,...,r (94) 4.2.5 Set of the residual generators Exploiting the model-based properties of state estimators the set of residual generators can be considered as *«*(*) = X flJk qjke(0 + YoJk(y(0 - D M u(0), fc = 1,2, . . ., m (95) Subsequently *ak(t) = X ak (T ak q(t) - ejk(O) + Y«jfcCq(0 = (X afc T afc + Y flJk C)q(0 - X, fc e fc (f) (96) To eliminate influences of the state variable vector it is necessary to consider X ak T ak + Y ak C = (97) X«*(I» " b uit (Cb Mfc ) el C) + Y flit C = (98) respectively. Choosing X ak = — C (98) gives - (C - Cb Mfc (Cb uit ) el C) + \ ak C = - (I m - Cb uit (Cb Mfc ) el )C + Y flit C = (99) i.e. \ ak = I m - Cb Hfc (Cb ujt ) el (100) Thus, the set of residuals (95) takes the form r«*(0 = (Im - Cb Hfc (Cb Hfc ) el )y(f) - Cq fe (f) (101) When all sensors are fault-free and a fault occurs in the Z-th actuator the residuals will satisfy the isolation logic IMOII < h sk , k = I, ||r sjk (*)|| > ^ * ^ / (102) This residual set can only isolate a single actuator fault at the same time. The principle can be generalized based on a regrouping of faults in such way that each residual will be designed to be sensitive to one group of actuator faults and insensitive to others. 322 Robust Control, Theory and Applications y 2 (t) - - / \ 1-" :f::::::::::|\::::| 1 j ; ; X : V y,(t) y 2 (t) " -f \ r I : V_ - \ \ \ ; - 1 V Fig. 4. System outputs for single actuator faults Fig. 5. Residuals for the 1st actuator fault Illustrative example Using the same system parameters as that given in the example in Subsection 4. 1.2 the next design parameters be computed b«i , (Cb wl ) el = [0.1333 0.0667] , T 0.8000 -0.3333 -0.1333 -0.4000 0.3333 -0.2667 -0.2000 -0.3333 0.8667 Hi (Cb w2 )^ [0.0862 0.0345" ifl2 0.6667 2.0000 0.3333 1.3333 2.0000 1.6667 -4.3333 -8.0000 -4.6667 , A 2 ifll 0.2 -0.4 -0.4 0.8 i fl 2 0.6379 -0.6207 -0.2586 -0.1207 0.7931 -0.0862 -0.6034 -1.0345 0.5690 1.2931 2.9655 0.6724^ 0.4310 0.6552 1.2241 -2.8448 -5.7241 -3.8793 0.1379 -0.3448 -0.3448 0.8621 Solving (86), (87) with respect to the LMI matrix variables P^, and Z^ using Design Principles of Active Robust Fault Tolerant Control Systems 323 f ~ \^ r/t) / — r 2 (t)- — ^ ; " 7Z.I1.ZI.1.3. r/t) [ 2 W f " ' Fig. 6. Residuals for the 2nd actuator fault Self-Dual-Minimization (SeDuMi) package for Matlab, the estimator gain matrix design problem was feasible with the results L al -a\ 0.0257 0.7321' 0.4346 0.2392 -0.7413 -0.7469 r«2 : ^2 0.2127 0.3382 -0.6686 0.9808' 0.0349 -0.4957 ,J2 0.7555 -0.0993 0.0619 0.0993 0.7464 0.1223 0.0619 0.1223 0.3920 _ 0.3504 1.2802" 0.9987 0.8810 _-2.2579 -2.3825_ , Li = 0.2247 0.7807 -2.8319 - 1.2173 0.7720 -2.6695 0.6768 -0.0702 0.0853" 0.0702 0.7617 0.0685 0.0853 0.0685 0.4637 _ 0.5888 1.6625" 0.6462 0.3270 -1.6457 -1.4233 , L 2 = 0.3878 0.6720 -2.6375 - 1.5821 0.3373 -1.8200 respectively. It is easily verified that the system matrices of state estimators are stable with the eigenvalue spectra p(T fl iA-JiC) = {-1.0000 - 1.6256 ± 0.3775 i} p(T fl2 A - J 2 C) = {-1.0000 - 1.5780 ± 0.4521 i} respectively, and the set of residuals takes the form r«i(0 Tali*) 0.2 -0.4 0.1379 -0.3448 -0.4 0.8 y(0 1 2 1 1 1 qie(0 -0.3448 0.8621 y(0- 1 2 1 1 1 q*(0 Fig. 4-6 plot the residuals variable trajectories over the duration of the system run. The results show that both residual profile show changes through the entire run, therefore a fault isolation has to be more sophisticated. 324 Robust Control, Theory and Applications 5. Control with virtual sensors 5.1 Stability of the system Considering a sensor fault then (1), (2) can be written as q / (0 = Aq / (f) + B M u / (0 (103) y f (t) = C f q f (t) + D u xi f (t) (104) where q/(t) G R n , xif(t) G K r are vectors of the state, and input variables of the faulty system, respectively, Qf G jrmxh ^ g ^ e ou tp U t ma trix of the system with a sensor fault, and Yf(t) G K m is a faulty measurement vector. This interpretation means that one row of Qf is null row. Problem of the interest is to design a stable closed-loop system with the output controller where u/(f) = -K y e (t) y e (f) = Ey/(f) + (C - EC/)q/ e (f) (105) (106) K G R rxm is the controller gain matrix, and E G R mxm is a switching matrix, generally used in such a way that E = 0, or E = l m . If E = full state vector estimation is used for control, if E = l m the outputs of the fault-free sensors are combined with the estimated state variables to substitute a missing output of the faulty sensor. Generally, the controller input is generated by the virtual sensor realized in the structure q fe (t) = Aq fe (t) + B u u f (t) + ](y f (t) - D u u f (t) - C f q fe (t)) (107) The main idea is, instead of adapting the controller to the faulty system virtually adapt the faulty system to the nominal controller. Theorem 3 Control of the faulty system with virtual sensor defined by (103) - (107) is stable in the sense of bounded real lemma if there exist positive definite symmetric matrices Q, R G R nxn , and matrices K G K rxm , J G R nxm such that 4>! QB M K (C- -EC/) -QB M K E (C/ - D M K (C - EC/) * 4>2 (D M K (C-EC/)) T -7 2 I r -(D u K E) f * * * * * — i-m <0 where <»! Q(A - B„K (C - EC/)) + (A - B„K (C - EC/)) T Q * 2 = R(A-JC/) + (A-JC/) T R Proof. Assembling (103), (104), and (107) gives q/(0 q/ e (0 A JC/A-JC/ q/(0 q/ e (0 B u B„ U/(f) y/(0 = C/q/M + D H u/(t) (108) (109) (110) (111) (112) Design Principles of Active Robust Fault Tolerant Control Systems 325 Thus, defining the estimation error vector vW = q/(0-q/e(0 as well as the congruence transform matrix T = T I I -I and then multiplying left-hand side of (111) by (114) results in q/(0 q/ e (0 A JC/A-JC/ r 1 T q f(0 >(0 + T B M A A - JC/ [ q/(0 " + B„" u /(f) u/(f) q/(0 respectively Subsequently, inserting (105), (106) into (116), (112) gives q/W together with A-B M K (C-EC/) B„K (C-EC/) A-JC, q/(0 + -B M K E y«(0 y/(f) = [CpD^^C-EC;) D M K (C-EC/)] q/W D„K Ey e (r) and it is evident, that the separation principle yields. Denoting qJW =[«!/(*) e^(0], w £ (f)=y e (f) ~A-B M K (C-EC / )B„K (C-EC / )1 [-B M K E A - JC/ \ ' £ [ C e =[C f - D„K (C - EC/) D„K (C - EC/)] , D e = -D M K E (121) To accept the separation principle a block diagonal symmetric matrix F £ > is chosen, i.e. (113) (114) (115) (116) (117) (118) (119) (120) P £ = diag[QR] where Q = Q T > 0, R = R T > 0, Q, R e R nxn Thus, with (109), (110) it yields 1 e-A-g + A £ 1 £ Oi QB W K (C-EC / ) * 4>2 P £ Bg QB W K E (122) (123) and inserting (121), (123), into (24) gives (108). This concludes the proof. ■ It is evident that there are the cross parameter interactions in the structure of (108). Since the separation principle pre-determines the estimator structure (error vectors are independent on the state as well as on the input variables), the controller, as well as estimator have to be designed independent. 326 Robust Control, Theory and Applications 5.2 Output feedback controller design Theorem 4 (Unified algebraic approach) A system (103), (104) with control law (105) is stable if there exist positive definite symmetric matrices P > 0, II = P 1 >0 such that B^(An + nA r )B^B^nc£ ■/« < 0, i = 0,1,2, ..., m (124) r»T± PA + A J P 7 2 I r r»T±T r'T± I« <0f = 1,2, ..., m, E = I„ C T1 (PA + A T P)C T1T C T1 Cj,. * — l m <0 f = 0,1,2, ... m, E = where n mT± (C-EC /z -) 7 E (125) (126) (127) and B^ is the orthogonal complement to B u . Then the control law gain matrix K exists if for obtained P there exist a symmetric matrices H > such that FHF T -e z FH + G;Kj * -H <0 (128) where i = 0,1,2, ... ,m, and ©i PA + A T P Cj- * -7 2 I r <0, F: PB M " r(c-Ec /f ) T l , G = E (129) Proof. Considering e<j(f) = then inserting Q = P (108) implies *! -PB„K E (C / -D„K (C-EC / )) J ,2 T ,~ .^f ' -7 Z Ir * (D„K E) ~*-m <0 (130) where 4>! = P(A - B M K (C - EC/)) + (A - B W K (C - EC f )) T F (131) For the simplicity it is considered in the next that D w = (in real physical systems this condition is satisfied) and subsequently (130), (131) can now be rewritten as PA + A 1 P "f PB W K [C-ECfEO] Yh * —I, (C-EC f ) T E (132) kJ[b£poo] <o Design Principles of Active Robust Fault Tolerant Control Systems 327 Defining the congruence transform matrix T^diagfp- 1 l r I w ] then pre-multiplying left-hand side and right-hand side of (132) by (133) gives (133) AP- 1 + P _1 A i P^C lr-Tn -7 2 I Since it yields K [(C-EC^P" 1 E B° B M * -I w P-^C-ECy) 7 E B^O I r l m (134) K o T [B^0 0] <0 (135) pre-multiplying left hand side of (134) by (135) as well as right-hand side of (134) by transposition of (135) leads to inequalities B^(AP" 1 + P- 1 A T )B^ JLp-lr-T B 7 |P ^C "7 2 Ir <0 (136) B^AP" 1 + P" 1 A T )B l [ T BjJ-P^Cj <0 (137) respectively Considering all possible structures Ca> i = 1, 2, . . . , m associated with simple sensor faults, as well as fault-free regime associated with the nominal matrix C = Cm, then using the substitution P _1 = II the inequality (136) implies (124). Analogously, using orthogonal complement -oT± (C-EC f ) T E (C-EC 7 ) 7 E In C* T± * lm (138) and pre-multiplying left-hand side of (132) by (138) and its right-hand side by transposition of (138) results in r »TJL PA + A T P * -7 2 I r r *T±_T r *T± C J <o (139) Considering all possible structures Ct\, i = 1,2, . . . , m (139) implies (125). Inequality (125) takes a simpler form if E = 0. Thus, now •^oTJL r c r- JL "C T± = I r 1^ (140) 328 Robust Control, Theory and Applications and pre-multiplying left-hand sides of (132) by (140) and its right-hand side by transposition of (140) results in r C T± (PA + A T P)C T±T C T± Cj" * -7 2 I r (141) which implies (126). This concludes the proof. ■ Solving LMI problem (124), (125), (126) with respect to LMI variable P, then it is possible to construct (128), and subsequently to solve (127) defining the feedback control gain K 0/ and H as LMI variables. Note, (124), (125), (126) have to be solved iteratively to obtain any approximation P _1 = II. This implies that these inequalities together define only the sufficient condition of a solution, and so one from (P, II -1 ) can be used in design independently while verifying solution using the latter. Since of an approximative solution the matrix defined in (129) need not be negative definite, and so it is necessary to introduce into (128) a negative definite matrix 9°r. as follows e^- = e /I --A<o (142) where A > 0. If (124), (125), (126) is infeasible the principle can be modified based on inequalities regrouping e.g. in such way that solving (124), (125), and (124), (126) separatively and obtaining two virtual sensor structures (one for E = and other for E = I m ). It is evident that virtual sensor switching be more sophisticated in this case. 5.3 Virtual sensor design Theorem 5 Virtual sensor (107) associated with the system (103), (104) is stable if there exist symmetric positive definite matrix R <G R nxn , and a matrix Z e R nxm / such that R = R J > RA + A T R - ZCfi + C^-Z T < The virtual sensor matrix parameter is then given as J = R J Z i = 0,1,2, ...,m (143) (144) (145) Proof. Supposing that q(t) = and D w = then (108), (110) is reduced as follows 4>2 * -7 2 I r <0 R(A-JC / ) + (A-JC / ) T R<0 respectively. Thus, with the notation Z = RJ (147) implies (144). This concludes the proof. (146) (147) (148) Design Principles of Active Robust Fault Tolerant Control Systems 329 Illustrative example Using for E = the same system parameters as that given in the example in Subsection 4.1.2 then the next design parameters were computed B [-0.8581 0.1907 0.4767] , C T± = [0.5774 -0.5774 0.5774] ~f0 1 2 1 1 1 , C '/l 000 1 1 , c /2 1 2 1 000 Solving (124) and the set of polytopic inequalities (126) with respect to P, II using the SeDuMi package the problem was feasible and the matrices 0.6836 0.0569 -0.0569 0.0569 0.6836 0.0569 -0.0569 0.0569 0.6836 as well as H = O.II2 was used to construct the next ones On 0.5688 0.9111 -3.0769 1 1 0.9111 -0.9100 -5.8103 2 1 3.0769 -5.8103 -6.7225 1 -0.1 1.0000 2.0000 1.0000 1.0000 1.0000 "0.7405 1.4810 0.7405 00 00" 1.8234 1.1386 3.3044 -0.1 0i, ©2 12 10000 1100000 To obtain negativity of 6°. the matrix A = 4.9417 was introduced. Solving the set of polytopic inequalities (128) with respect to K the problem was also feasible and it gave the result K -0.0734 -0.0008 -0.1292 0.1307 which secure robustness of control stability with respect to all structures of output matrices Qfi, i = 0, 1, 2. In this sense p(A-B u K C) = [-1.0000 -1.3941 ± 2.3919 i] p(A-B u K C fl ) = [-1.0000 -2.2603 ± 1.6601 i] p(A-B u K C f2 ) = [-1.0000 -1.1337 ± 1.8591 i] Solving the set of polytopic inequalities (144) with respect to R, Z the feasible solution was R 0.7188 0.0010 0.0016 0.0010 0.7212 0.0448 0.0016 0.0448 0.1299 -0.0006 0.4457 0.0117 0.0701 -0.0629 -0.5894 Thus, the virtual sensor gain matrix J was computed as 0.0002 0.6296 0.0473 0.3868 -0.5003 -4.6799 330 Robust Control, Theory and Applications y,(t) y 2 (t)) /\ ....: \T V T / ^ y e1 (t) y e2 w) f I V / t 1 : : - 10 15 20 25 30 Fig. 7. System output and its estimation which secure robustness of virtual sensor stability with respect to all structures of output matrices C a-, i = 0, 1, 2. In this sense p(A-JC) -1.0000 -1.1656 -3.4455 KA-JC/i) -1.0000 -1.2760 -3.7405 p(A-B u K C f2 ) -1.0000 -1.1337+ 1.8591 i -1.1337- 1.8591 i As was mentioned above the simulation results were obtained by solving the semi-definite programming problem under Matlab with SeDuMi package 1.2, where the initial conditions were set to q(0) = [0.2 0.2 0.2] T , q,(0) = [0 0] T respectively, and the control law in forced mode was u f (t) = -K y e (t)+w(t), w(*)= [-0.2-0.2] 7 Fig. 7 shows the trajectory of the system outputs and the trajectory of the estimate system outputs using virtual sensor structure. It can be seen there a reaction time available to perform fault detection and isolation in the trajectory of the estimate system outputs, as well as a reaction time of control system reconfiguration in the system output trajectory. The results confirm that the true signals and their estimation always reside between limits given by static system error of the closed-loop structure. 6. Active control structures with a single actuator fault 6.1 Stability of the system Theorem 6 Fault tolerant control system defined by (1) - (9) is stable in the sense of bounded real lemma if there exist positive definite symmetric matrices Q, R <G R nxn , S <G R lxl , and matrices Design Principles of Active Robust Fault Tolerant Control Systems 331 k e R rxn , l e R rx/ , J e R nxm , M e R lxL , n e R 0> n QB W K QB W L * a> 22 R(B / - JD / ) - (SNC) T * * 3>33 ?!xifi m such that (C-D„K) T (D H K) r SM S (D H L) r 7 2 Il * — 7 2 I/ * * — Im <0 (149) where 0> n = Q(A - B M K) + (A - B W K) T Q, <D 22 = R(A - JC) + (A - JC) T R (150) a> 33 = S(M - ND / ) + (M - ND / ) T S (151) Proof. Considering equality i(t) = i(t) and assembling this equality with (1) - (4), and with (7) - (9) gives the result rq(01 *(t) f(0 [te(t)\ B M K B 7 B„L JC A-B U K-JC JD 7 B f -JV f -B u L NC -NC ND f M-ND y= [C-D H KD / -D„L] q e (0 ie{t) Tq(01 " o " q*(0 f( f ) + f(0 Lf,wJ (152) which can be written in a compact form as q a (0 = A a q a (f) + f a (0 y = C«q a (f) where qj(0 = [q T (0 q, T (0 f T (0 #(')] , «I(0 = [o T o r F(t) o r ] (153) (154) (155) (156) (157) (158) (159) where e<j(f) is the error between the actual state and the estimated state, and eAt) is the error between the actual fault and the estimated fault, respectively then it is possible to define the state transformation An B W K B / B W L JC A-B M K-JC JD f B / -JD / -B W L NC -NC ND '/ M-ND / C« = [C -D w KD r D M L] Using notations e,(0 = q(0-q*(0/ e/Ct) = f(t) - f e (f) q/»(0 = Tq»(t) "q(0' f(f) e/(0 ,f^(f)=Tf„(f) i(t) m , T I I -I 10 I -I (160) 332 Robust Control, Theory and Applications and to rewrite (154), (155) as follows qp(0=A /J q /J (t)+f/j(0 y = Cpqp(0 where TA a T A B M K B f -B U L B U L A - JC B f - JD f -NC -M M-ND /J C^ = CJ" 1 = [ C - D W K D W K D f - D W L D W L ] Since (5) implies B 7 - B W L = 0, V f - D M L = it obvious that (163), (165) can be simplified as Aa = TA^T" 1 (161) (162) (163) (164) (165) (166) (167) (168) (169) (170) (171) (172) To apply the separation principle a block diagonal symmetric matrix P^ > has to be chosen, A B W K B M L A - JC Bf-JDf -NC -MM-NDf C^ = CJ" 1 = [C - D M K D W K D M L] Eliminating out equality i(t) = i(t) it can be written q*(0 = A,q*(0+B*w*(0 y = C s qs(t) + D 3 w 3 (t) where qj(0 = [q T (0 <W ej(t)] , wj(t) = [f{t) '?{t)\ A B M K B W L A - JC B / - JD / -NC M-ND /J 0" -M I C s = [C-D M KD„KD H L], D,= [0 0] P^ = diag[QRS] where Q, R 6 R nxn , S 6 R lx} . Thus, with (150), (151) it yields FgAs + AiVs On QB U K QB„L * *22 R(B / -JD / )- (SNC) T * * *33 ,V S B S 0" -SM S (173) (174) and inserting (171), (172), and (174) into (24) gives (149). This concludes the proof. Design Principles of Active Robust Fault Tolerant Control Systems 333 6.2 Feedback controller gain matrix design It is evident that there are the cross parameter interactions in the structure of (149). Since the separation principle pre-determines the estimator structure (error vectors are independent on the state as well as on the input variables), at the first design step can be computed a feedback controller gain matrix K, and at the next step be designed the estimators gain matrices J G R nxm , M e R /x/ , N e R lxm , including obtained K. Theorem 7. For a fault-free system (1), (2) exists a stable nominal control (4) if there exist a positive definite symmetric matrix X > 0, X <G R nxn , a matrix Y £ R rxn , and a positive scalar 7 > 0, 7 <G R such that X AX + XA T - Y T B 7 T X T >0 B M Y B W L XC T -Y T D£ -7 2 Il L T D£ * — i-m <0 The control law gain matrix is then given as K = YX _1 Proof. Considering e^(f) = 0, then separating q(f) from (168)-(169) gives q(f)=A°q(t) + B°w°(f) y(0 = C°q(f) + D°w°(f) where w°(t) = e / (0 A° = A - B„K, B° = B U L, C° = C - D U K, and with (181), and P = Q inequality (24) can be written as D° D U L TnTl QA + A T Q - QB U K - K t b£q QB u L C t - K j D - 7 2 i ; L T r>l <0 Introducing the congruence transform matrix H^diagfQ" 1 Ijl m ] then multiplying left-hand side, as well right-hand side of (182) by (183) gives AQ 1 + Q 1 A i -bjkq^-qicb; b m l qh^-k^d :Tj^T\ -Im <0 With notation X > 0, KQ _1 (175) (176) (177) (178) (179) (180) (181) (182) (183) (184) (185) (184) implies (176). This concludes the proof. 334 Robust Control, Theory and Applications 6.3 Estimator system matrix design Theorem 8 For given scalar j > 0, j e R, and matrices Q = Q T > 0, Q e R nxn , K e R rxn , L G R rxl estimators (7) - (9) associated with the system (1), (2) are stable if there exist symmetric positive definite matrices R e R nxn , S e K /x/ , and matrices Z e R nxm , V e R lxl , W e R lxl ?lxm such that R S O22 RB '/■ ZDy R T > S T >0 (WC) T 33 -V - 7 2 I/ s -7 2 Il (D„K) r (D H L) r Am where ■ C T Z T , o 33 O22 = RA - ZC + A J R The estimators matrix parameters are then given as M = S _1 V, N = S _1 W, J = R" ] Proof. Supposing that q(f) = then (149) is reduced as follows V- WD f + V T -DTW T r (186) (187) (188) (189) (190) <D 22 R(B / - JD / ) - (SNC) T -SM 3>33 -7 2 I/ s -7 2 Il (D W K) T (D W L) T — Am <o (191) where «D 22 = R(A-JC) + (A-JC) T R / Thus, with notation <D 33 S(M - ND / ) + (M - ND / ) T S (192) SM = V, SN = W, RJ = Z (193) (191), (192) implies (188), (189). This concludes the proof. It is obvious that V e = A — JC, as well as M have to be stable matrices. 6.4 Illustrative example To demonstrate algorithm properties it was assumed that the system is given by (1), (2) where 1 0" 1 Vu = "1 3" 2 1 Bf = "1" 2 L = "-1" 5 -9 -5_ 15 1 C = "1 1 2 1" 1 , D, 1 — 00" 00 > »J "0" Design Principles of Active Robust Fault Tolerant Control Systems 335 f(t) 1 ft : : V I f 10 15 20 25 30 15 20 25 30 35 40 t[s] Fig. 8. The first actuator fault as well as its estimation and system input variables f y e1 w y e2 W) - Hi-- : If ; 20 25 30 35 Fig. 9. System output and its estimation Solving (175), (176) with respect to the LMI matrix variables 7, X, and Y using Self-Dual-Minimization (SeDuMi) package for Matlab, the feedback gain matrix design problem was feasible with the result I.7454 -0.8739 0.0393 -0.8739 1.3075 -0.5109 0.0393 -0.5109 2.0436 0.9591 1.2907-0.1049 -0.1950 -0.5166 -0.4480 7 = 1.8509 K 1.2524 1.7652 0.0436 -0.0488 -0.2624 -0.3428 In the next step the solution to (186) - (188) using design parameters 7 = 1.8509 was also feasible giving the LMI variables V = -1.3690, S = 1.1307, W = [0.9831 0.7989] R 1.7475 0.0013 0.0128 0.0013 1.4330 0.0709 0.0128 0.0709 0.6918 -0.0320 1.0384 0.1972 0.1420 -2.0509 -1.1577 336 Robust Control, Theory and Applications which gives 0.0035 0.6066 0.2857 0.1828 -2.9938 -1.7033 N 0.8694 0.7066 M: -1.2108 Since M < it is evident that the fault estimator is stable and verifying the rest subsystem stability it can see that ^qe A-B M K A-JC -1.1062 0.0282 0.9847 -2.4561 -3.2659 1.2555 -6.0087 -9.4430 -3.3297_ -0.6101 0.3864 -0.0035" -0.4684 -0.7541 0.7143 -0.3029-1.3092-2.0062 {-0.7110 - 3.4954 ±i 4.3387} Q(A qe ) = {-1.0000 - 1.1852 ±i 0.7328} where q(-) is eigenvalue spectrum of a real square matrix. It is evident that the designed observer-based control structure results the stable system. The example is shown of the closed-loop system response in the autonomous mode where Fig. 8 represents the first actuator fault as well as its estimation, and the system input variables, respectively, and Fig. 9 is concerned with the system outputs and its estimation, respectively. 7. Concluding remarks This chapter provides an introduction to the aspects of reconfigurable control design method with emphasis on the stability conditions and related system properties. Presented viewpoint has been that non-expansive system properties formulated in the Hoo design conditions underpins the nature of dynamic and feedback properties. Sufficient conditions of asymptotic stability of systems have thus been central to this approach. Obtained closed-loop eigenvalues express the internal dynamics of the system and they are directly related to aspects of system performance as well as affected by the different types of faults. On the other hand, control structures alternation achieved under virtual sensors, or by design or re-design of an actuator fault estimation can be done robust with respect of unaccepted faults. The role and significance of another reconfiguration principles may be found e.g. in the literature (Blanke et al.,2003), (Krokavec and Filasova,2007), (Noura et al.,2009), and references therein. 8. Acknowledgment The main results presented in this chapter was derived from works supported by VEGA, Grant Agency of Ministry of Education and Academy of Science of Slovak Republic, under Grant No. 1/0256/11. This support is very gratefully acknowledged. 9. References [Blanke et al.,2003] Blanke, M.; Kinnaert, M; Lunze, J. & Staroswiecki, M. (2003). Diagnosis and Fault-tolerant Control Springer, ISBN 3-540-01056-4, Berlin. [Boyd et al.,1994] Boyd, D.; El Ghaoui, L.; Peron, E. & and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM Society for Industrial and Applied Mathematics, ISBN, 0-89871-334-X, Philadelphia Design Principles of Active Robust Fault Tolerant Control Systems 337 [Chen and Patton,1999] Chen, J. & Patton, R.J. (1999). Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers, ISBN 0-7923-8411-3, Norwell. [Chen et al.,1999] Chen, J.; Patton, R.J. & Chen, Z. (1999). Active fault-tolerant flight control systems design using the linear matrix inequality method. Transactions of the Institute of Measurement and Control, ol. 21, No. 2, (1999), pp. 77-84, ISSN 0142-3312. [Chiang et al.,2001] Chiang, L.H.; Russell, E.L. & Braatz, R.D. (2001). Fault Detection and Diagnosis in Industrial Systems. Springer, ISBN 1-85233-327-8, London. [Ding,2008] Ding, S.X. (2008). Model-based Fault Diagnosis Techniques: Design Schemes, Alghorithms, and Tools. Springer, ISBN 978-3-540-76304-8, Berlin. [Dong et al.,2009] Dong, Q.; Zhong, M. & Ding, S.X. (2009). On active fault tolerant control for a class of time-delay systems. Preprints of 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, SAFEPROCESS 2009, pp. 882-886, Barcelona, Spain, June 30, - July 3, 2009. [Ducard,2009] Ducard, G.J.J. (2009). Fault-tolerant Flight Control and Guidance Systems. Practical Methods for Small Unmanned Aerial Vehicles. Springer, ISBN 978-1-84882-560-4, London. [Filasova and Krokavec,2009] Filasova, A. & Krokavec, D. (2009). LMI-supported design of residual generators based on unknown-input estimator scheme. Preprints ot the 6 th IFAC Symposium on Robust Control Design ROCOND '09, pp. 313-319, Haifa, Israel, June 16-18, 2009, [Gahinet et al.,1995] P. Gahinet, P.; Nemirovski, A.; Laub, A.J. & Chilali, M. (1995). EMI Control Toolbox User's Guide, The Math Works, Natick. [Herrmann et al.,2007] Herrmann, G.; Turner, M.C. & Postlethwaite, I. (2007). Linear matrix inequalities in control. Mathematical Methods for Robust and Nonlinear Control, Turner, M.C. and Bates, D.G. (Eds.), pp. 123-142, Springer, ISBN 978-1-84800-024-7, Berlin. [Jiang,2005] Jiang, J. (2005). Fault-tolerant Control Systems. An Introductory Overview. Acta Automatica Sinica, Vol. 31, No. 1, (2005), pp. 161-174, ISSN 0254-4156. [Krokavec and Filasova,2007] Krokavec, D. & Filasova, A. (2007). Dynamic Systems Diagnosis. Elfa, ISBN 978-80-8086-060-8, Kosice (in Slovak). [Krokavec and Filasova,2008] Krokavec, D. & Filasova, A. (2008) Diagnostics and reconfiguration of control systems. Advances in Electrical and Electronic Engineering, Vol. 7, No. 1-2, (2008), pp. 15-20, ISSN 1336-1376. [Krokavec and Filasova,2008] Krokavec D. & A. Filasova, A. (2008). Performance of reconfiguration structures based on the constrained control. Proceedigs of the 17 th IFAC World Congress 2008, pp. 1243-1248, Seoul, Korea, July 06-11, 2008. [Krokavec and Filasova,2009] Krokavec, D. & Filasova, A. (2009). Control reconfiguration based on the constrained LQ control algorithms. Preprints of 7 th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes SAFEPROCESS 2009, pp. 686-691, Barcelona, Spain, June 30, - July 3, 2009. [Liao et al.,2002] Liao, E; Wang, J.L. & Yang, G.H. (2002). Reliable robust flight tracking control. An LMI approach. IEEE Transactions on Control Systems Technology, Vol. 10, No. 1, (2002), pp. 76-89, ISSN 1063-6536. [Nesterov and Nemirovsky,1994] Nesterov, Y; & Nemirovsky, A. (1994). Interior Point Polynomial Methods in Convex Programming. Theory and Applications, SIAM, ISBN 0-89871-319-6, Philadelphia. [Nobrega et al.,2000] Nobrega, E.G.; Abdalla, M.O. & Grigoriadis, K.M. (2000). LMI-based filter design for fault detection and isolation. Proceedings of the 39 th IEEE Conference Decision and Control 2000, Vol. 5, pp. 4329-4334, Sydney, Australia, December 12-15, 2000. 338 Robust Control, Theory and Applications [Noura et al.,2009] Noura, H.; Theilliol, D.; Ponsart, J.C. & Chamseddine, A. (2009). Fault-tolerant Control Systems. Design and Practical Applications. Springer, ISBN 978-1-84882-652-6, Berlin. [Patton,1997] Patton. R.J. (1997). Fault-tolerant control. The 1997 situation. Proceedings of the 3 rd IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes SAFEPROCESSS97, Vol. 2, pp. 1033-1054, Hull, England, August 26-28, 1997. [Peaucelle et al.,2002] Peaucelle, D.; Henrion, D.; Labit, Y. & Taitz, K. (2002). User's Guide for Sedumi Interface 1.04. LAAS-CNRS, Toulouse. [Simani et al.,2003] Simani, S.; Fantuzzi, C. & Patton, R.J. (2003). Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques. Springer, ISBN 1-85233-685-4, London. [Skelton et al.,1998] Skelton, R.E., Iwasaki, T. & Grigoriadis, K. (1998) A Unified Algebraic Approach to Linear Control Design. Taylor & Francis, ISBN 0-7484-0592-5, London. [Staroswiecki,2005] Staroswiecki, M. (2005). Fault tolerant control. The pseudo-inverse method revisited. Proceedings of the 16 th IFAC World Congress 2005, Prag, Czech Repulic, July 4-8, 2005. [Theilliol et al.2008] Theilliol, D.; Join, C. & Zhang, Y.M. (2008). Actuator fault tolerant control design based on a reconfigurable reference input. International Journal of Applied Mathematics and Computer Science, Vol.18, No.4, (2008), pp. 553-560, ISSN 1641-876X. [Zhang and Jiang,2008] Zhang, Y.M. & Jiang, J. (2008). Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in Control, Vol. 32, (2008), pp. 229U-252, ISSN 1367-5788. [Zhou and Ren,2001] Zhou, K.M. & Ren, Z. (2001). A new controller architecture for high performance, robust and fault tolerant control. IEEE Transactions on Automatic Control, Vol. 40, No. 10, (2001), pp. 1613-1618, ISSN 0018-9286. 15 Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control Alejandro H. Gonzalez 1 and Darci Odloak 2 institute of Technological Development for the Chemical Industry (INTEC), CONICET - Universidad Nacional del Litoral (U.N.L.). Giiemes 3450, (3000) Santa Fe, department of Chemical Engineering, University of Sao Paulo, Av. Prof. Luciano Gualberto, trv 3 380, 61548 Sao Paulo, Argentina 2 Brazil 1. Introduction Model Predictive Control (MPC) is frequently implemented as one of the layers of a control structure where a Real Time Optimization (RTO) algorithm - laying in an upper layer of this structure - defines optimal targets for some of the inputs and/ or outputs (Kassmann et al., 2000). The main scope is to reach the most profitable operation of the process system while preserving safety and product specification constraints. The model predictive controller is expected to drive the plant to the optimal operating point, while minimizing the dynamic error along the input and output paths. Since in the control structure considered here the model predictive controller is designed to track the optimal targets, it is expected that for nonlinear process systems, the linear model included in the controller will become uncertain as we move from the design condition to the optimal condition. The robust MPC presented in this chapter explicitly accounts for model uncertainty of open loop stable systems, where a different model corresponds to each operating point of the process system. In this way, even in the presence of model uncertainty, the controller is capable of maintaining all outputs within feasible zones, while reaching the desired optimal targets. In several other process systems, the aim of the MPC layer is not to guide all the controlled variables to optimal targets, but only to maintain them inside appropriate ranges or zones. This strategy is designated as zone control (Maciejowski, 2002). The zone control may be adopted in some systems, where there are highly correlated outputs to be controlled, and there are not enough inputs to control all the outputs. Another class of zone control problems relates to using the surge capacity of tanks to smooth out the operation of a process unit. In this case, it is desired to let the level of the tank to float between limits, as necessary, to buffer disturbances between sections of a plant. The paper by Qin and Badgwell (2003), which surveys the existing industrial MPC technology, describes a variety of industrial controllers and mention that they always provide a zone control option. Other example of zone control can be found in Zanin et al, (2002), where the authors exemplify the application of this 340 Robust Control, Theory and Applications strategy in the real time optimization of a FCC system. Although this strategy shows to have an acceptable performance, stability is not usually proved, even when an infinite horizon is used, since the control system keeps switching from one controller to another throughout the continuous operation of the process. There are several research works that treat the problem of how to obtain a stable MPC with fixed output set points. Although stability of the closed loop is commonly achieved by means of an infinite prediction horizon, the problem of how to eliminate output steady state offset when a supervisory layer produces optimal economic set points, and how to explicitly incorporate the model uncertainty into the control problem formulation for this case, remain an open issue. For the nominal model case, Rawlings (2000), Pannochia and Rawlings (2003), Muske and Badgwell (2002), show how to include disturbance models in order to assure that the inputs and states are led to the desired values without offset. Muske and Badgwell (2002) and Pannochia and Rawlings (2003) develop rank conditions to assure the detectability of the augmented model. For the uncertain system, Odloak (2004) develops a robust MPC for the multi-plant uncertainty (that is, for a finite set of possible models) that uses a non-increasing cost constraint (Badgwell, 1997). In this strategy, the MPC cost function to be minimized is computed using a nominal model, but the non-increasing cost constraint is settled for each of the models belonging to the set. The stability is then achieved by means of the recursive feasibility of the optimization problem, instead of the optimality. On the other hand, there exist some recent MPC formulations that are based on the existence of a control Lyapunov function (CLF), which is independent of the control cost function. Although the construction of the CFL may not be a trivial task, these formulations also allow the explicit characterization of the stability region subject to constraints and they do not need an infinite output horizon. Mashkar et al. (2006) explore this approach for the control of nominal nonlinear systems, and Mashkar (2006) extends the approach for the case of model uncertainty and control actuator fault. More recently, Gonzalez et al. (2009) extended the infinite horizon approach to stabilize the closed loop with the MPC controller for the case of multi-model uncertainty and optimizing targets. They developed a robust MPC by adapting the non-increasing cost constraint strategy to the case of zone control of the outputs and it is desirable to guide some of the manipulated inputs to the targets given by a supervisory stationary optimization stage, while maintaining the controlled output in their corresponding zones, taking into account a finite set of possible models. This problem, that seems to interchange an output tracking by an input-tracking formulation, is not trivial, since once the output lies outside the corresponding zone (because of a disturbance, or a change in the output zones), the priority of the controller is again to control the outputs, even if this implies that the input must be settled apart from its targets. Since in many process systems, mainly from the chemical and petrochemical industries, the process model shows significant time delays, the main contribution of this chapter is to extend the approach of Gonzalez et al. (2009) to the case of input delayed multi-model systems by introducing minor modifications in the state space model, in such a way that the structure of the control algorithm is preserved. Simulation of a process system of the oil refining industry illustrates the performance of the proposed strategy. 2. System representation Consider a system with nu inputs and ny outputs, and assume for simplicity that the poles relating any input Uj to any output y; are non-repeated. To account for the implementation of Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 341 an intuitive MPC formulation, an output prediction oriented model (OPOM) originally presented in Odloak (2004) is adapted here to the case of time delayed systems. Let us designate # f the time delay between input Uj and output y\, and define p > max# • Then, the state space model considered here is defined as follows: l, i x(k + l) = Ax(k) + BAu(k) y(k) = Cx(k) where x(k) = [y(k) T y(k + l) T ••• y(k + p) T x s (k) T x d (k) T J (1) U • • - S, ~ ny • S 2 • 1-ny , B = S v • • ny 5F((? + i)r) Sp+i • • m J B s • • F B d C = V«y X s g K ny , x d g C nd , Fe C n •] (2) W g K nymd , I ny = diag([l • • • 1]) g K nyxn v . The advantage of using the structure of the transition matrix A is that the state vector is divided into components that are associated to the system modes. In the state equation (1), the state components x s correspond to the (predicted) output steady state, which are in addition the integrating modes of the system (the integrating modes are induced by the incremental form of the inputs), and the components x d correspond to the stable modes of the system. Naturally, when the system approaches steady state these last components tend to zero. For the case of non-repeated pole, F is a diagonal matrix with components of the form e nT where r\ is a pole of the system and T is the sampling period. It is assumed that the system has nd stable poles and B s is the gain matrix of the system. The upper left block of matrix A is included to account for the time delay of the system. Si, ... , S v +\ are the step response coefficients of the system. Matrix *F , which appears in the extended state matrix, is defined as follows nt)- Mt) o ... o o Mt) - o o o where m = [e r >^'-^ - eWM-4) M) r inul (t-O inu ) a i, mi A V i, nn I r inuna (t-a «], 342 Robust Control, Theory and Applications r if : k , with k=l,...,na, are the poles of the transfer function that relates input Uj and output \ji and na is the order of this transfer function. It is assumed that na is the same for any pair (uj, \)i). The time delay affects the dimension of the state matrix A through parameter p and the components of matrix W . Input matrix B is also affected by the value of the time delay as the step response coefficients S n will be equal to zero for any n smaller than the time delay. 2.1 Model uncertainty With the model structure presented in (1), model uncertainty is related to uncertainty in matrices F, B s , B d and the matrix of time delays 6 . The uncertainty in these parameters also reflects in the uncertainty of the step response coefficients, which appear in (2). There are several practical ways to represent model uncertainty in model predictive control. One simple way to represent model uncertainty is to consider the multi-plant system (Badgwell, 1997), where we have a discrete set Q of plants, and the real plant is unknown, but it is assumed to be one of the components of this set. With this representation of model uncertainty, we can define the set of possible plants as Q = {& 1 ,...,& L } where each © n corresponds to a particular plant: © n = (f, B s , B d ,6\ , n = 1, ..., L . Also, let us assume that the true plant, which lies within the set Q is designated as <9r and there is a most likely plant that also lies in Q and is designated as <9 N . In addition, it is assumed that the current estimated state corresponds to the true plant. Badgwell (1997) developed a robust linear quadratic regulator for stable systems with the multi-plant uncertainty. Later, Odloak (2004) extended the method of Badgwell to the output tracking of stable systems considering the same kind of model uncertainty. These strategies include a new constraint corresponding to each of the models lying in Q, that prevents an increase in the true plant cost function at successive time steps. More recently, Gonzalez and Odloak (2009) presented an extension of the method by combining the approach presented in Odloak (2004) with the idea of including the output set point as a new restricted optimization variable to develop a robust MPC for systems where the control objective is to maintain the outputs into their corresponding feasible zone, while reaching the desired optimal input target given by the supervisory stationary optimization. In this work the controller proposed by Gonzalez et al. (2009) is extended to the case of uncertain systems with time delays. 2.2. System steady state As was already said, one of the advantages of the model defined in (1) and (2) is that the state component x s (k) represents the predicted output at steady state, and furthermore this component concentrates the integrating modes of the system. Observe that for the model defined in (1) and (2), if Au(k + ;) = for ; > , then the future states can be computed as follows x(k + j) = A j x(k) Assuming that F has all the eigenvalues inside the unit circle (i.e. the system is open loop stable), it is easy to show that Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 343 limA ; : Then, it becomes clear that Km x(k + j) = \. "0 • • l ny °1 - l «y - l ny ° ••• ^=r X s (kf ... : s (k) T o] and consequently limy (A: + /') = Climx(A: + ;') = x s (k) • Therefore, x s (k) can be interpreted as the prediction of the ;->oo ;->oo output at steady state. The state component x s (k) is assumed to be known or estimated through a stable state observer. A stable observer for the model defined in (1) and (2) is given by x(k + l\k + l) = x(k + l\k) + K(y(k + l)-Cx(k + l\k)) where K = I • • • I is the observer gain, and x(k + l\k + l) = x(k + l\k) + K(y(k + l)-Cx(k + l\k)) x(k + l\k + l) = (I-KC)Ax(k\k) + (I-KC)BAu(k) For open loop stable systems this is a stable observer as matrix (I-KC)A has the eigenvalues of F and the remaining eigenvalues are equal to zero. 3. Control structure In this work, we consider the control structure shown in Figure 1. In this structure, the economic optimization stage is dedicated to the calculation of the (stationary) desired target, u desk , for the input manipulated variables. This stage may be based on a rigorous stationary model and takes into account the process measurements and some economic parameters. In addition, this stage works with a smaller frequency than the low-level control stage, which allows a separation between the two stages. In the zone control framework the low-level control stage, given by the MPC controller, is devoted to guide the manipulated input from the current stationary value u ss to the desired value given by the supervisory economic stage, u desk , while keeping the outputs within specified zones. In general, the target UdesM will vary whenever the plant operation or the economic parameters change. If it is assumed that the system is currently at a stationary value given by (u ss ,y ss ), the desired target Ud eS/ k should satisfy not only the input constraints W min - U des,k - W max but also the output zone condition (3) 344 Robust Control, Theory and Applications where u min and w max represent the lower and upper bounds of the input, y min and y max represent the lower and upper limits of the output, B s (0 n ) is the gain corresponding to a given model n , and x s n (k) is the estimated steady-state values of the output corresponding to model n . Note that in the control structure depicted in Figure 1, as the model structure adopted here has integral action, the estimation of component x s n (k) tends to the measured output at steady state for all the models lying in Q, which means that x s n (k) = y ss if the system is at steady state (See Gonzalez and Odloak 2009 for details). Taking into account this fact, equation (3) can be rewritten as ymin^ BS {®n) U des,k +d n,ss^yn n = l,-,L, (4) where d n/SS = x s n (k) - B s (® n )u ss = y ss - B s (<5> n )u ss is the output bias based on the comparison between the current actual output at steady state and the current predicted output at steady state for each model. In other words, B s (0 )u d k + d can be interpreted as the corrected k _ output steady state. Note that, since u ss =y\Auij) , for a large k, the term B s (0 n )u ss 7=0 represents the output prediction based only on the past inputs. economic i r parameter 5 Economic Optimization m 4 input and i output range: I if IIiTIi ■' if IlLtT t Rjobust MFC ^ w t~~ } £u(k) \{k) m System measured Observer disturbances unmeasured -l{k) disturbances Observer a — Fig. 1. Control structure. Based on the later concepts, it is possible to define two input feasible sets for the stationary desired target Ud es ,h The first one is the global input feasible set S = [u : w min < u < w max } , which represents a box-type set. In addition, it is possible to define the more restricted input feasible set & u , which is computed taking into account both, the input constraints and the output limits: Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 345 and y min < B s (<9 M ) u - B s (& n )u ss + y ss < y n (5) This set, which depends on the current stationary point given by ( u ss ,y ss ), is the intersection of several sets, each one corresponding to a model lying in set Q. When the output zones are narrow, the restricted input feasible set is smaller than the global feasible set, defined solely by the input constraints. An intuitive diagram of the input feasible set is shown in Figure 4, where three models are used to represent the uncertainty set. In the following sections it will be shown that the proposed controller remains stable and feasible even when the desired input target u deS/k is outside the set S u , or the set S u itself is null. 4. Nominal MPC with zone control and input target One way to handle the zone control strategy, that is, to maintain the controlled output inside its corresponding range, is by means of an appropriate choice of the output error penalization in the conventional MPC cost function. In this case the output weight is made equal to zero when the system output is inside the range, and the output weight is different from zero if the output prediction is violating any of the constraints, so that the output variable is strictly controlled only if it is outside the feasible range. In this way, the closed loop is guided to a feasible steady state. In Zanin et al. (2002), an algorithm assigns three possible values to the output set points used in the MPC controller: the upper bound of the output feasible range if the predicted output is larger than the upper bound; the lower bound of the output feasible range if the predicted output is smaller than this lower bound; and the predicted output itself, if the predicted output is inside the feasible range. However, a rigorous analysis of the stability of this strategy is not possible even when using an infinite output horizon. Gonzalez et al. (2006) describe a stable MPC based on the incremental model defined in (1) and (2), that takes into account a stationary optimization of the plant operation. The controller was designed specifically for a heat exchanger network with a number of degrees of freedom larger than zero. In that work, the mismatch between the stationary and the dynamic model was treated by means of an appropriate choice of the weighting matrices in the control cost. However, stability and offset elimination was assured only when the model was perfect. Based on the work of Gonzalez et al (2006), we consider the following nominal cost function: Vk = Z {(y(* + i I k ) - Vs V ,k ) Q y {y( k + i\ k )- Vs V ,k ) + (*(* + / 1 *) - *w ) T Q u (6) m-1 (u(k + j\k)-u deS/k j\+ZAu(k + j\k) T RAu(k + j/k) j=Q where Au(k + j\k) is the control move computed at time k to be applied at time k+j, m is the control or input horizon, Q ,Q U ,R are positive weighting matrices of appropriate dimension, y sp/ k and Ud es ,k are the output and input targets, respectively. The output target y $v ,k becomes a computed set point when the output has no optimizing target and consequently the output is controlled by zone. This cost explicitly incorporates an input deviation penalty that tries to accommodate the system at an optimal economic stationary point. 346 Robust Control, Theory and Applications In the case of systems without time delay the term corresponding to the infinite output error in the cost Vk is divided in two parts: the first goes from the current time k to the end of the control horizon, k+m-1; while the second one goes from time k+m to infinity. This is so because beyond the control horizon no control actions are implemented and so, considering only the state at time k+m, the infinite series can be reduced to a single terminal cost. In the case of time delayed systems, however, the horizon beyond which the entire output evolution can be predicted by a terminal cost is given by k+p. As a result, the cost defined in (6) can be developed as follows v k = I {(y(* + /' I *) - Vs P ,k ) Q y (y(* + ; I *) - y„,* ) ;=0 CO J + T\{y( k+ p + J\ k )-ys P ,k) Q y {y(k+p+j\k)-y sp , k ) ;=0 (u(k + j/k)- u deSfk ) Q u (u(k + j/k)- u deS/k )} + J] zlw(fc + j I k) T RAu(k + j / k) ;=0 The first term on the right hand side of (7) can be developed as follows v k,i = Z {{y( k + ; I fc ) - Vfc ) Q y (y(* + ; I fc ) - y sp ^ ) (7) where v k,i ={yk- ~ l v y sp ,k ) Q y {yk - i y y sp ,k ) Vk'- y k =N x x(k) + SAu k V(k\k) 1 y(fc + l|fc) . N x =[l (p+1)nv 0] 6 «C +1 )"^;S = y(fc + p|fc)_ (8) s 2 s 1 s p s p-l c J p-m+l I =[l ••• I T J e<R (p+1); 111/ X IT}/ Q y = diag Qy Qy p+1 nx = (p + l)ny + ny + nd Consequently, considering (8), the term V kl can be written as follows Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 347 \l = [X*W + SAu k - ^sr^ Qy [M x x(k) + SAu k - I y y sprk ] (9) The term corresponding to the infinite horizon error on the system output in (7) can be written as follows OO ry v kr2 = 1 E{y( k+ p + i\ k )-y S p,k) Q y {y(k+p+j\k)-y sp , k ) OO ry V kil = Y J {x\k + m\k) + ^{ V + j-m)x d {k + m\k)-y SVik ) Q y 7=1 (x s (k + m\k) + *F(p + j-m)x d (k + tn\k)-y spfk ) where, x s (k + m\k) = x s (k) + B s Au k ,B S = B s ••• B s (10) Also, Au k =\Au(k/kf ••• Au(k + m-l/kfJ eK mMU x\k + m\k) = F m x\k) + B d Au k ,B d = \F m - 1 B d F m ~ 2 B d ••• B° W{p + j-m) = Y(p - m)P (11) In order to force V k ,i to be bounded, we include the following constraint in the control problem x s (k + m | k) - y SVrk = or x s (k) + B s Au k - y sp/k = With the above equation and (11), Eq. (10) becomes V k/2 =^(*F(p-m)F j x d (k + m\ky) Q y (w(p-m)F j x d (k + m\k) ) j 7=1 V k>2 = f [ F m x d (k) + B d Au k ) T Q d (F m x d (k) + B d Au k ) where Q d =T(np-™)F j ) Q^ np - m )FJ) 7=1 Finally, the infinite term corresponding to the error on the input along the infinite horizon in (7) can be written as follows V k,3=H{ U ( k + J\ k )- U des,k) Qu{ U ( k + J\ k )- U des,k) 7=1 (12) 348 Robust Control, Theory and Applications Then, it is clear that in order to force (12) to be bounded one needs the inclusion of the following constraint u(k + m\k)-u deS/k =0 <k-l) + ~I T u Au k -u des>k =0 (13) where J T = ••• L Then, assuming that (13) is satisfied, (12) can be written as follows Vk,3 = (*««(* " 1) + MAu k - I u u deSrk f Q u (l u u(k - 1) + MAu k - I u u deSrk ) nu • •• nu nu •• ; Qu = diag Qu Qu nu nu nu V 111 where M - Now, taking into account the proposed terminal constraints, the control cost defined in (7) can be written as follows V k = [N x x(k) + SAu k - I y y SV/k ] Q y [N x x(k) + SAu k - I y y sp/k _ +(F m x d (k) + B d Au k f Q d (F m x d (k) + B d Au k ) +( I u u{k - 1) + MAu k - I u u deS/k f Q u [l u u{k - 1) + MAu k - I u u deS/k ) + Au T k RAu k . To formulate the IHMPC with zone control and input target for the time delayed nominal system, it is convenient to consider the output set point as an additional decision variable of the control problem and the controller results from the solution to the following optimization problem: min V k = Au[HAu k + 2c T f Au k A u k ,y spik subject to "(fc-i) + ^")t-"<fea = o (14) x s (k) + &Au k -y sp)k =0 (15) ymin — Jsp,k — i/max (16) ax <Au(k + j\k)<Au max 7 = 0,1,- • ,ra-l <u(k-l) + ^Au(k + i\k)<u max ; 7 = 0,1,. z=0 • ,m-\ Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 349 where H = S T Q y S + B dT Q d B d + M T Q U M + R c] = x(k) T N x T Q y S + x d (k) T (F m ) T Q d B d + (u(k - 1) - u des f I T U Q U M Constraints (14) and (15) are terminal constraints, and they mean that both, the input and the integrating component of the output errors will be null at the end of the control horizon m. Constraint (16), on the other hand, forces the new decision variable y $v ,k to be inside the zone given by y m i n and y max . So, as y sv ,k is a set point variable, constraint (16) means that the effective output set point of the proposed controller is now the complete feasible zone. Notice that if the output bounds are settled so that the upper bound equals the lower bound, then the problem becomes the traditional set point tracking problem. 4.1 Enlarging the feasible region The set of constraints added to the optimization problem in the last section may produce a severe reduction in the feasible region of the resulting controller. Specifically, since the input increments are usually bounded, the terminal constraints frequently result in infeasible problems, which means that it is not possible for the controller to achieve the constraints in m time steps, given that m is frequently small to reduce the computational cost. A possible solution to this problem is to incorporate slack variables in the terminal constraints. So, assuming that the slack variables are unconstrained, it is possible to guarantee that the control problem will be feasible. Besides, these slack variables must be penalized in the cost function with large weights to assure the constraint violation will be minimized by the control actions. Thus, the cost function can be written as follows v k V T -■T,{y^ + j\^)-y S p f k- s y f k) Q y {y{ k + i\ k )-ys V ,k- 5 y,k) ;=0 CO J- +Y l {y( k+ p + i\ k )-ys?,k- s y,k) Q y (y{ k +p+i\ k )-y sp ,k-Sy, k )+ m-1 T + Ys{ U ( k + J\ k )- U des,k-<5 u ,k) Qu{u(k + J\k)-Udes,k-<5u,k) ^ GO +^(u(k + m + j\k)-u deS/k -S U/k ) Q u (u(k + m + j\k)-u deSfk -S U/k ) + 7=0 ra-l + X M* + ; I k) T RAu(k + j\k) + s T yik s y s yik + s u /s u s Utk where S , S u are positive definite matrices of appropriate dimension and S k <Eyi ny , S Ufk e$i nu are the slack variables (new decision variables) that eliminate any infeasibility of the control problem. Following the same steps as in the controller where slacks are not considered, it can be shown that the cost defined in (17) will be bounded if the following constraints are included in the control problem: x s {k) + B s Au k -y SVrk -S yrk =0 350 Robust Control, Theory and Applications u{k-l) + I l u Au k - Ude$ik -S Uik =0 (18) In this case, the cost defined in (17) can be reduced to the following quadratic function where v k = [ Au l yl v ,k 5 ]m 5 l,\ H n H 12 H 13 H 14 H 21 H 22 H 23 H 31 H 32 H 33 H 41 H 44 *y,* S u,k +2 [ c /4 C /,2 c /,3 C / J y s? a '■'/i ; /i' H u = S T Q V S + (B d ) T Q d B d + M T Q U M + R H 12 = H 2 T a = -S T Q y I y , H 13 = H 3 T a = -S T Q y I y , H 14 = H 4 T a = -M T Q J M H 22 = *« QA ' H 23 = H 32 = l y Qy l y ' H 33 = l y Qy l y + S y / H 44 = *u QA + S u H 24 = H 42 = H 34 = H 43 = Tit a c f/1 =x(k) T N T x Q v S + x d (k) T (F m ) T Q d B d m +(u(k-l)-u des y I T U Q U M c fi2 = -x(k) T N T x Q± , c f * = -x(k) T N T x QJ., y y c fA = -(u(k - 1) - u deS/k ) llQ u I u c = x(k) T NlQ y NXk) + x d (k) T (F m ) T Q d F m x d (k) + (u(k-l) I T u Q u I u (u(k-l)-u deS/k ) Then, the nominally stable MPC controller with guaranteed feasibility for the case of output zone control of time delayed systems with input targets results from the solution to the following optimization problem: Problem PI min V k Au k'Vs V ,k' 5 y,k' 5 u,k subject to: -Au max <Au(k + j\k)<Au n j = 0,l,—,m-l Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 351 w min ^-l) + £zlu(/c + z|/c)<u max ; j = Q,l,'~,m-l j I ymm^ysp,k^ymax ( 19 ) x s (k) + B s Au k -y spfk -S ¥fk =0 (x s (k + m\k)-y spfk -S ¥/k =0) u(k - 1) + I T U Au k - u deS/k - S U/k = (u(k + m-l\k)- u deS/k - 8 u>k = 0) It must be noted that the use of slack variables is not only convenient to avoid dynamic feasibility problems, but also to prevent stationary feasibility problems. Stationary feasibility problems are usually produced by the supervisory optimization level shown in the control structure defined in Figure 1. In such a case, for instance, the slack variable 8 k allows the predicted output to be different from the set point variable y k at steady state (notice that only y k is constrained to be inside the desired zone). So, the slacked problem formulation allows the system output to remain outside the desired zone, if no stationary feasible solution can be found. It can be shown that the controller produced through the solution of problem PI results in a stable closed loop system for the nominal system. However, the aim here is to extend this formulation to the case of multi model uncertainty. 5. Robust MPC with zone control and input target In the model formulation presented in (1) and (2) for the time delayed system, uncertainty concentrates not only on matrices F, B s and B d as in the system without time delay, but also on matrix e yi n y xnu that contains all the time delays between the system inputs and outputs. Observe that the step response coefficients Si,...,S p +i, which appears in the input matrix and ^(p + 1), which appears in the state matrix of the model defined in (1) and (2) are also uncertain, but can be computed from F, B s , B d and 6 . Now, considering the multi- model uncertainty, assume that each model is designated by a set of parameters defined as © n = \B s n ,B n ,¥ n ,6 n \ , n = l,...,L. Also, assume that in this case p >max# n (z,;) + ra (this condition guarantees that the state vector of all models have the same dimension). Then, for each model 6> , we can define a cost function as follows ;=0 + T,{yn( k + P + i\ k )-y S p,k( n)- S y,k( n)) Qy{y n ( k + V + J \ k ) -y S p,k( n) ~ ^y,k{ n)) m-\ j, + ^(u(k + j\k)- u deS/k - S U/k ) Q u (u(k + j\k)- u deS/k - S U/k ) ;=0 00 + ^(u(k + m + j\k)-u deSrk -S U/k ) Q u (u(k + m + j\k)-u deS/k -S U/k ) ;=0 m-\ + X Au{k + j | k) T RAu(k + j\k) + Sl k (0 n )S y S yik (0 n ) + 8jS u 8 uM (20) 352 Robust Control, Theory and Applications Following the same steps as in case of the nominal system, we can conclude that the cost defined in (20) will be bounded if the control actions, set points and slack variables are such that (18) is satisfied and x s (k) + B s (0 n )Au k - yspik (0 n )-S yik (0 n ) = O Then, if these conditions are satisfied, (20) can be written as follows M&n) = {NXV + S(®„)Au k - I y y sp , k {0 n )-l y 8 yfk {0 n )f Q y (N x x(k) + ~S(@n)Au k - l y y sv , k {0 n ) - I y S yM {0 n )) + (F(0Xx\k) + Bi(0 n )Au k fQ d (0 n )(F(0 n ) m x d (k) + Bi(0 n )Au k ) (21) + (l«"(fc " 1) + MAu k - I u u deS/k - I u S Uik ) Q u (!„«(* - 1) + MAu k - I u u deSfk - I u 8 U/k ) + Au T k RAu k + S yM {0 n ) T S y S y , k {0 n ) + 8jS u 8 Ufk V k (&„) = [M y T SV:k {0 n ) 8] ik {0 n ) 8 T Uil H n (0 n ) H 12 (0„) H 13 (0 n ) H u H 2l( n) H ll W 23 ° ^31 (<§>„) H 32 W 33 ° ■2[c /4 K) L f,2 L f,3 L fA AUp Vs Vf k{ n) #y,k(®n) J u,k H A -C(&n) H A Auu y sp A n) ty, k (®n) iJ n,)< H 11 (© n )=~S(©jQ~S(© n ) + (B\© n )) T Q d (©^ H 12 =Hi 1 =-S(© n ) 1 QJ / H 13 =H> 1 =-S(© n ) 1 Q v I v , H^Hi^-M 1 Q U I U c v ' y TV H 22 = l y Qy l y > H 23 = H 32 = l y Qy l y > H 33 = l y Qy l y H 24 = H 42 = H 34 = H 43 = c f/1 = x(k) T N T x Q y S(© n ) + x d (kf(F(© n r fQ xd B\© n ) + (u(k - 1) - u des f I T U Q U M \Txtt; \T^rTr c f/2 = -x(ky N x Q y I y , c f/3 = -x(kY N x Q y I y c fA =-(u(k-l)-u des fl T u Q u I u Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 353 c = x{k?N T x Q y N x x{k) + x d {k) T {¥{0 n ) m ) T Q xd ¥{0 n )"'x d {k) + -(u(k - 1) - u deSik f 1 T U QJ U (u(k - 1) - u deSik ) Then, the robust MPC for the system with time delay and multi-model uncertainty is obtained from the solution to the following problem: Problem P2 AUk>y 8 p,k( & n)> S y,k( & n)> S u* n=l,...,L subject to "^max < M k + j\ k )^ ^max J = 0,l,~',m-l i Wmin^^fc-lJ + X^^ + ^I^^Wmax; 7 = 0,1,-,"* -1 ymm^ y S p,k( n)^y ma x', n = i,---,L x s (k) + B s (0 n )Au k -y sp/k (0 n )-S y/k (0 n ) = O; n = l,-,L (23) u(k-l) + llAu k -u deSfk -S Ufk =0 (24) Vk(^k>8y,k{®n)A,k>y8p,^ n = l,...,L (25) where, assuming that (zl^_ 1 ,y sprfc _ 1 (6> n ),^^_ 1 ,^ fc _ 1 (6> n )) is the optimal solution to Problem P2 at time step k-1, we define Au k =[Au(k\k-l) T ». zl M *(fc + m-2|fc-l) T 0] T ; jf^(© II ) = y^ l _ 1 (© II ) and ^ such that «(* - 1) + F u Au k - UdesM - 8 Utk = (26) and define 5 k {0 n ) such that x s (k) + B s (0 n )Au k -y sp:k (0 n )-S V)k (0 n ) = O (27) In (20), <9 N corresponds to the nominal or most probable model of the system. Remark 1: The cost to be minimized in problem P2 corresponds to the nominal model. However, constraints (23) and (24) are imposed considering the estimated state of each model n e Q . Constraint (25) is a non-increasing cost constraint that assures the convergence of the true state cost to zero. Remark 2: The introduction of L set-point variables allows the simultaneous zeroing of all the output slack variables. In that case, whenever possible, the set-point variable y sPfk (® n ) 354 Robust Control, Theory and Applications will be equal to the output prediction at steady state (represented by x s n (k + m) ), and so the corresponding output penalization will be removed from the cost. As a result, the controller gains some flexibility that allows achieving the other control objectives. Remark 3: Note that by hypothesis, one of the observers is based on the actual plant model, and if the initial and the final steady states are known, then the estimated state x T (k) will be equal to the actual plant state at each time k. Remark 4: Conditions (26) and (27) are used to update the pseudo variables of constraint (25), by taking into account the current state estimation x s n (k) for each of the models lying in Q , and the last value of the input target. One important feature that should have a constrained controller is the recursive feasibility (i.e. if the optimization problem is feasible at a given time step, it should remain feasible at any subsequent time step). The following lemma shows how the proposed controller achieves this property. Lemma. If problem P2 is feasible at time step k, it will remain feasible at any subsequent time step k+j, 7=1,2, . . . Proof: Assume that the output zones remain fixed, and also assume that Au k =lAu(k\kf ••• Au(k + m-l\kfJ e^ mMU , (28) yl v A®i)>---,ys V A®L)> ^(6> 1 ),-,^(6> L ) and S* Uik (29) correspond to the optimal solution to problem P2 at time k. Consider now the pseudo variables (Au k+1/ y spM1 (0 1 ) / --- / y spM1 (& L ) / ^ +1 (6> 1 ),..., Au k+1 =lAu(k + l\kf ... Au(k + m-l\kf ol (30) y S p,k + i(®n) = yl P ,k(®n)> n = l,---,L, (31) Also, the slacks S U/k+1 and S y/k+1 (0 n ) are such that u(k) + I T U Au k+1 - u deSfk - § uM1 = (32) and x s n (k + l) + B s (0 n )Au k+1 -y spM1 (0 n )-S V/k+1 {0 n ) = O, n = \ L (33) We can show that the solution defined through (30) to (33) represent a feasible solution to problem P2 at time k+1, which proves the recursive feasibility. This means that if problem P2 is feasible at time step k, then, it will remain feasible at all the successive time steps k+1, fe+2 / ...D Now, the convergence of the closed loop system with the robust controller resulting from the later optimization problem can be stated as follows: Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 355 Theorem. Suppose that the undisturbed system starts at a known steady state and one of the state observers is based on the actual model of the plant. Consider also that the input target is moved to a new value, or the boundaries of the output zones are modified. Then, if condition (3) is satisfied for each model n e C2 , the cost function of the undisturbed true system in closed loop with the controller defined through the solution to problem P2 will converge to zero. Proof: Suppose that, at time k the uncertain system starts from a steady state corresponding to output y(k) = y ss and input u(k-l) = u ss . We have already shown that, with the model structure considered in (1) and (2), the model states corresponding to this initial steady state can be represented as follows: *»(*) = Vss ■•• Vss Vss ° , n = l,-,L V and consequently, x s n {k) = y ss , x„(k) = 0, n = l,-~,L. At time k, the cost corresponding to the solution defined in (28) and (29) for the true model is given by CO J 1 Vk{^) = U(yT(k + j\k)-y sM (& T )-Sl k (0 T )) QjyAk + jm-y^i^-S;^)) + (u\k + j\k)-u deSik -5' Uik ) Q u (u(k + j\k)-u deS/k -5* Urk )} (34) m-1 + ^Au\k + j\k) T RAu\k + j\k) + S;/(0 T )S/ y/k {0 T ) + S:/sX /k 7=0 At time step k+1, the cost corresponding to the pseudo variables defined in (30) to (33) for the true model is given by Z\(yT(k+j+i\k)-y; Prk (0 T )-s; rk (0 T )) Q y (i T (k + j + i\k)- y ; Prk (0 T )-s; rk (0 T )) + (u* (k + j + 1 / k)-u deS/k -S* U/k ) Q u (u\k + j + l/k)-u deS/k -Sl /k )} m-\ 7=0 Observe that, since the same input sequence is used and the current estimated state corresponding to the actual model of the plant is equal to the actual state, then the predicted state and output trajectory will be the same as the optimal predicted trajectories at time step k. That is, for any j ; > 1 , we have x T (k + j\k + l) = x T (k + j\ k) 356 Robust Control, Theory and Applications and y T (k + j\k + l) = y T (k + j\k) In addition, for the true model we have 8 k+1 {© T ) = 8 k (& T ) and 8 U/k+1 = S U/k . However, the first of these equalities is not true for the other models, as for these models we have x n (k + l\k + l)*x n (k + l\k), for <9 n *<9 T . Now, subtracting (35) from (34) we have Vk (*r ) - % + i (®t ) = {yr(k\k)- y SVfk (6> T ) - 8] M (6> T )f Q y (y T (k\k)- y sv>k (<9 T ) - 8] >k (6> T )) +{u\k\k)-u deSik -8* Uik ) Q u (u\k\k)-u deSfk -S* Ufk ) + Au(k) T RAu(k) and, from constraint (25), the following relation is obtained 4K)^ W (0 T ), which finally implies Vk (@t ) " Vii (®r )>{y r (k\k)- y ; a (<9 T ) - Sl k (0 T )) T Q y (y T {k\k)- y\ ?x (0 T ) - S* yX (0 r )) +(u\k\k)-u deSik -S' U/k ) Q u (u(k\k)-u ieS/k -S* U/k ) + Au(k) T RAu{k) (36) Since the right hand side of (36) is positive definite, the successive values of the cost will be strictly decreasing and for a large enough time k , we will have \V^{© T )-V^ +1 {0 T j\ = O , which proves the convergence of the cost. The convergence of V k (& T ) means that, at steady state, the following relations should hold At steady state, the state is such that zto*(fc) = y(*) yik) y(k) o t n (fc)= y( k +p) x s n (k) K{k) where y(k) is the actual plant output. Note that the state component x n (k) is null as it corresponds to the stable modes of the system and the input increment is null at steady state. Then, constraint (23) can be written as follows: yik) Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 357 s lk( n)=yn{k\k)-y sp r k ( n) = y{k)-y sp r k (0n), n=i,...,i. (37) This means that, if the output of the true system is stabilized inside the output zone, then the set point corresponding to each particular model will be placed by the optimizer exactly at the output predicted values. As a result, all the output slacks will be null. On the other hand, if the output of the true system is stabilized at a value outside the output zone, then the set-point variable corresponding to any particular model will be placed by the optimizer at the boundary of the zone. In this case, the output slack variables will be different from zero, but they will all have the same numerical value as can be seen from (37). Now, to strictly prove the convergence of the input and output to their corresponding targets, we must show that slacks S u t and 8 k (®j) will converge to zero. It is necessary at this point to notice that in the case of zone control the degrees of freedom of the system are no longer the same as in the fixed set-point problem. So, the desired input values may be exactly achieved by the true system, even in the presence of some bounded disturbances. Let us now assume that the system is stabilized at a point where, 8 k -(& 1 ) = --- = S ^(<9 L )^0, and S u t ± . In addition, assume that the desired input value is constant at u deSfk . Then, at time k large enough, the cost corresponding to model n will be reduced to ^ft) = <i^)VM( »i) + <M,P n = l,...,L, (38) and constraints (21) and (22) become, xW)-y sp rk( «) = S yA & «)> n = l,-,L (39) and u{k-l)-u desrk =8 urk . Since x s n (k) = y(k), n = l,---,L ,Eq. (39) can be written as Now, we want to show that if u(k -l) and u des k are not on the boundary of the input operating range, then it is possible to guide the system toward a point in which the slack variables S k (0 n ) and S U/k are null, and this point have a smaller cost than the steady state defined above. Assume also for simplicity that m=l. Let us consider a candidate solution to problem P2 defined by: Au(k/k) = u desk --u(k-l) = -S uk - (40) and ys P A n)=y{k)-v s (0nK- k »-i l («) Now, consider the cost function defined in (21), written for time step k and the control move defined in (40) and the output set point defined in (41): 358 Robust Control, Theory and Applications %{& n ) = (i y y(k) - s,(&„Kt - i y y sp ,-k( n)- h s v ,- k ( n)) Qy (l y y(k) - S x {& n )8 uX -l y y spX {0 n )- IyS yX (0 n )) + {F(0j t x\k)-B\0 n )S u ^QA0 n ){H0„) m x i {k)-B i {0 n )S uM ) + {i„ u ( k - 1) - MS u,k ~ iuU ies ,k - 'A,* ) Q U ( J«"( fc - 1) - ms u1 - - I u u deS/k - l u s U/k ) -(-<? V ) T R(S A + S T (0 n ) T S v 5 V (0„) + S t T S u S t v u,k ' v u,k ' y,k \ n/ y y,fc \ «/ «,fc u u,k Now, since the solution defined by \Au(k/k),S t{& ),S ?-] satisfies constraint (23) and (24), the above cost can be reduced to V- k (®n) = €,An(&nK k - where S^ in (& n ) = p y F (0„ ) - S, (0„ )J Q y [l y B s («9„ ) - S, (<9„ )] + B d (0 n f Q d B d (0 n ) + R Then, if S„>Sr((9„), n = l,..,L, (42) the cost corresponding to the decision variables defined in (40) and (41) will be smaller than the cost obtained in (38). This means that it is not possible for the system to remain at a point in which the slack variables 5 k (& n ) , n = l,'--,L and S U/k are different from zero. Thus, as long as the system remains controllable, condition (42) is sufficient to guarantee the convergence of the system inputs to their target while the system output will remain within the output zones. □ Observe that only matrix S u is involved in condition (42) because condition (3) assures that the corrected output prediction, i.e. the one corresponding to the desired input values, lies in the feasible zone. In this case, for all positive matrices S y , the total cost can be reduced by making the set point variable equal to the steady-state output prediction, which is a feasible solution and produces no additional cost. However, matrix S y is suggested to be large enough to avoid any numerical problem in the optimization solution. Remark 5: We can prove the stability of the proposed zone controller under the same assumptions considered in the proof of the convergence. Output tracking stability means that for every />0 , there exists a p(y) such that if |x T (0)| < p , then |jc r (/c)|</ for all k > ; where the extended state of the true system x T (k) may be defined as follows x T (k) = y T (k\k)-x s T (k) y T (k + p\k)-x s T (k) *r( k )-y^-i(®r) 4(k) Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 359 To simplify the proof, we still assume that m=l, and suppose that the optimal solution obtained at step k-l is given by Au k _ 1 = Au (k-l / k-l) , yl p , k - 1 (®i),''',y* 8p , k -i(@L)> s ]m-\{ \)''"' 5 ]m-\{ l) and ^_i • A feasible solution to problem P2 at time k is given by: M k = ° / Vs V ,k (®n ) = ¥*sp,k-l (®n ) / and K,k and $y,k {®n ) are such that u{k-l) + I T u Au k -u deSik -S Uik =Q (43) ^W + B s (0„)^-y sM (<9„)-^(0„) = O, n = \ L. (44) Since Au(k \ k) = , we have u(/c | /c) = u(/c - 1) and from (43) we can write Kk= u i k \ k )- U des,k (45) For the true system, (44) can be written as follows 4(k)-yl P , k -i(0 T )-s yrk (0 T ) = o and consequently, we have the following relations ^(<S > r) = 4W-y S M-iK) (46) and m = Ss P ,k-i(®r) + £ v A T) (47) For the feasible solution defined above, the cost defined in (21) can be written for the actual model T as follows + (F((9 T rx^W) T Q x ,((9 T )(F((9 T rx^(fc)) + (l u u(k - 1) - I u u deS/k - I u S U/k ) Q u (l u u(k - 1) - I u u deSfk - I u S U/k ) Now, using (45), (46) and (47) the cost defined in (48) can be reduced to the following expression V k (&r) = x T (k) T jc[Q y C a + Cj (F(<9 T f f Q d (<9 T )(F(<9 T f )c 2 + C 3 T S y C 3 + C 4 T S„Q where 1 — I (p+l)ny (p+l)nyxny (p+l)nyxnd (p+l)nyxnuj 360 Robust Control, Theory and Applications ^2 [yndx(p+l)ny ^ndxny *nd "ndxnu \ 3 I nyx(p+l)ny ny nyxnd nyxnu } Q - \_^nux(p+l)ny ® nuxny ^nuxnd Thus, the cost defined in (48) can be written as follows: Inn] V 2/k (0 T ) = x T (k) T H 1 (0 T )x T (k), (49) where H a = C T 1 Q y C 1 +C T 2 (F(0 T ) m f Q xd (0 T )(F(0 T ) m )c 2 + C T 3 Sf 3 + ClS u C 4 . Because of constraint (25), the optimal true cost (that is, the cost based on the true model, considering the optimal solution that minimizes the nominal cost at time k) will satisfy v;(0 T )<v k (0 T ). (50) and V* + »(»r)*V*(»r)foranyn>l. (51) By a similar procedure as above and based on the optimal solution at time k+n, we can find a feasible solution to Problem P2 at time k + n + 1, for any w>l, such that { {0 T )<Vl n {0 T ) (52) and from the definition of K we have fWi(«r) = »r (* + » + !) H,(0 T )x T (k + n + l) Therefore, combining inequalities (49) to (52) results x T (k + n + lf H 1 (0 T )x T (k + n + l)<x T (kf H 1 (0 T )x T (k), Vn>l. As H a (<9 T ) is positive definite, it follows that \k + n + 1)| < a (6> T )|x T (fc)|, Vn > 1 where i{0 T )- 4™( H i( t)) 1/2 < max -UxJHi^-))' ^in(Hl(0;)) 1/2 If we restrict the state at time k to the set defined by hc T (fc)| < /? , then, the state at tine k+n+1 will be inside the set defined by Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 361 p T (k + n + l)\\<a(0 T )p, Vn>l . Which proves stability of the closed loop system, as x T will remain inside the ball |x T | < a{© T )p , where oc(® T ) is limited, as long as the closed loop starts from a state inside the ball \\x T \\< P ■ Therefore, as we have already proved the convergence of the closed loop, we can now assure that under the assumption of state controllability at the final equilibrium point, the proposed MPC is asymptotically stable.D Remark 6: It is important to observe that even if condition (3) cannot be satisfied by the input target, or the input target is such that one or more outputs need to be kept outside their zones, the proposed controller will still be stable. This is a consequence of the decreasing property of the cost function (inequality (36)) and the inclusion of appropriate slack variables into the optimization problem. When no feasible solution exists, the system will evolve to an operating point in which the slack variables, which at steady state are the same for all the models, are as small as possible, but different from zero. This is an important aspect of the controller, as in practical applications a disturbance may move the system to a point from which it is not possible to reach a steady state that satisfies (3). When this happens, the controller will do the best to compensate the disturbance, while maintaining the system under control. Remark 7: We may consider the case when the desired input target u deS/k is outside the feasible set & u and the case where the set & u itself is null. If & u is not null, the input target Udes,k could be located within the global input feasible set & , but outside the restricted input feasible set & u . In this case, the slack variables at steady state, S U/SS and 8 ss (<9 n ) , cannot be simultaneously zeroed, and the relative magnitude of matrices S y and S u will define the equilibrium point. If the priority is to maintain the output inside the corresponding range, the choice must be S » S u , while preserving S u > S™ . Then, the controller will guide the system to a point in which 8 ss (<9 n ) « 0, n = l,---,L and 8 U/SS * . On the other hand, if 3 U is null, that is, there is no input belonging to the global input feasible set S that simultaneously satisfies all the zones for the models lying in Q , then, the slack variables 8 ss {O n ), n = l,'--,L , cannot be zeroed, no matter the value of 8 U/SS . In this case (assuming that S y »S U ), the slack variables 8 y /SS (@ n ),n = 1,--,L, will be made as small as possible, independently of the value of 8 U/SS . Then, once the output slack is established, the input slack will be accommodated to satisfy these values of the outputs. 6. Simulation results for the system with time delay The system adopted to test the performance of the robust controller presented here is based on the FCC system presented in Sotomayor and Odloak (2005) and Gonzalez et al. (2009). It is a typical example of the chemical process industry, and instead of output set points, this system has output zones. The objective of the controller is then to guide the manipulated inputs to the corresponding targets and to maintain the outputs (that are more numerous than the inputs) within the corresponding feasible zones. The system considered here has 2 inputs and 3 outputs. Three models constitute the multi-model set Q on which the robust controller is based. In two of these models, time delays were included to represent a possible degradation of the process conditions along an operation campaign. The third model corresponds to the process at the design conditions. The parameters corresponding to each of these models can be seen in the following transfer functions: 362 Robust Control, Theory and Applications G{© 1 ) = 0.4515 e~ ls 2.9846s + 1 1.5e -6 s 0.20336 1.7187s + 1 (0.1886s- 3.8087) e~ 3s 20s + 1 17.7347s 2 + 10.8348s + 1 1.7455 e~ bs 9.1085s + 1 -6.1355 e^ s 10.9088s + 1 G(0 2 )- 0.25 e~ IS 0.135 e~ bs 3.5s + 1 2.77s + 1 (0.1886s- 2.8) g" 4s 25s + 1 19.7347s 2 + 10.8348s + 1 0.9e" 1.25 e -5s 11.1085s + 1 -5e~ 12.9088s + 1 G{0 3 ) = 0.7 1.98s + 1 2.3 25s + 1 3 0.5 2.7s + 1 0.1886s -4.8087 15.7347s 2 + 10.8348s + 1 -8.1355 7s + l 7.9088s + 1 In this reduced system, the manipulated input variables correspond to: u\ air flow rate to the catalyst regenerator, m opening of the regenerated catalyst valve, and the controlled outputs are the following: \j\ riser temperature, 1/2 regenerator dense phase temperature, 1/3: regenerator dilute phase temperature. In the simulations considered here, model 1 is assumed to be the true model, while model <9 3 represents the nominal model that is used into the MPC cost. In the discussion that follows, unless explicitly mentioned, the adopted tuning parameters of the controller are m -- Si, :1(T T = l, Q y =0.5* diag(l 1 1), Q u =diag(l l), R = diag(l l), v * diag(l 1 1) and S u = 10 5 * diag(l l) . The input and output constraints, as well as the maximum input increments, are shown in Tables 1 and 2. Output 1/min l/max yi (°c) y 2 (°c) ys(°Q 510 600 530 530 610 590 Table 1. Output zones of the FCC system Input Au max Wmin Wmax Mi (ton/h) 25 w 2 (%) 25 75 25 250 101 Table 2. Input constraints of the FCC system Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 363 Before starting the detailed analysis of the properties of the proposed robust controller, we find it useful to justify the need for a robust controller for this specific system. We compare, the performance of the proposed robust controller defined through Problem P2, with the performance of the nominal MPC defined through Problem PI. We consider the same scenario described above except for the input targets that are not fully included in the control problem (we consider a target only to input u\ by simply making Q u = diag(l 0) and S u = 10 5 * diag(l 0) . This is a possible situation that may happen in practice when the process optimizer is sending a target to one of the outputs. Figures 2 and 3 show the output and input responses respectively for the two controllers when the system starts from a steady state where the outputs are outside their zones. It is clear that the conventional MPC cannot stabilize the plant corresponding to model 1 when the controller uses model <9 3 to calculate the output predictions. However, the proposed robust controller performs quite well and is able to bring the three outputs to their zones 550 r 500 L 800 r 10 15 20 25 30 35 40 45 time (min) 50 >, 600 400 L 10 15 20 25 30 time (min) 35 40 45 50 1000 r CO 500 10 15 20 25 30 time (min) 35 40 45 50 Fig. 2. Controlled outputs for the nominal ( ) and robust ( ) MPC. We now concentrate our analysis on the application of the proposed controller to the FCC system. As was defined in Eq. (5), each of the three models produces an input feasible set, whose intersection constitutes the restricted input feasible set of the controller. These sets have different shapes and sizes for different stationary operating points (since the disturbance d n (k) is included into Eq. (5), except for the true model case, where the input feasible set remains unmodified as the estimated states exactly match the true states. The closed loop simulation begins at u ss = [230.5977 60.2359] and y ss = [549.5011 704.2756 690.6233], which are values taken from the real FCC system. For such an operating point, the input feasible set corresponding to models 1, 2 and 3 are depicted in Figure 4. These sets are quite distinct from each other, which results in an empty restricted feasible input set for the controller ( & u = & u (& 1 ) fl & u (& 2 ) H $ u (^3 ) )• This means that, we cannot find an input that, 364 Robust Control, Theory and Applications taking into account the gains of all the models and all the estimated states, satisfies the output constraints. 250, ^ ^ ^ ^ ^ < < < ^ , 200 150 L 5 10 15 20 25 30 35 40 45 50 time (min) 80 - r \ / \ I / - - I I f - / \ ' / \ <N 60 > V \ ' \ — ~ / \ =5 40 on / \ ~r \ / \ i iii 5 10 15 20 25 30 35 40 45 50 time (min) Fig. 3. Manipulated inputs for the nominal ( ) and robust ( ) MPC. 80 Fig. 4. Input feasible sets of the FCC system Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 365 The first objective of the control simulation is to stabilize the system input at u des - [165 60] . This input corresponds to the output y = [520 606.8 577.6] for the true system (& 1 ) , which results in the input feasible sets shown in Figure 5a. In this figure, it can be seen that the input feasible set corresponding to model 1 is the same as in Fig. 4, while the sets corresponding to the other models adapt their shape and size to the new steady state. Once the system is stabilized at this new steady state, we simulate a step change in the target of the input (at time step k=50 min). The new target is given by u b des = [175 64] , and the corresponding input feasible sets are shown in Figure 5b. In this case, it can be seen that the new target remains inside the new input feasible set 3 U , which means that the cost can be guided to zero for the true model. Finally, at time step k=100 min, when the system reaches the steady state, a different input target is introduced (u c des =[175 58] ). Differently from the previous targets, this new target is outside the input feasible set 3 C U , as can be seen in Figure 5c. Since in this case, the cost cannot be guided to zero and the output requirements are more important than the input ones, the inputs are stabilized in a feasible point as close as possible to the desired target. This is an interesting property of the controller as such a change in the target is likely to occur in the real plant operation. KW S\ ^v ■ «L J)y * /A£ u w - V"*-^. ~K{d 2 ) 120 130 140 150 160 170 180 190 200 210 120 130 140 150 160 170 181 u1 u1 190 200 210 Fig. 5. (a): Initial input feasible sets; (b): Input feasible sets when the first input target is changed; and (c): Input feasible sets when the second input target is changed. 366 Robust Control, Theory and Applications 550 \ V^^ >, 500 ( D 50 time (min) 100 150 700 <N 650 - V - 600 _/L ( 700 <2> 600 3 50 time (min) 100 150 1_. >> i \y^~ Rnn 50 100 150 time (min) Fig. 6. Controlled outputs and set points for the FCC subsystem with modified input target. 240 220 \ - 200 \ 180 \ 160 ^ 50 100 time (min) 150 T 7CH \ 2 60 50 L 50 100 150 time (min) Fig. 7. Manipulated inputs for the FCC subsystem with different input target. Figure 6 shows the true system outputs (solid line), the set point variables (dotted line) and the output zones (dashed line) for the complete sequence of changes. Figure 7, on the other hand, shows the inputs (solid line), and the input targets (dotted line) for the same sequence. As was established in Theorem 1, the cost function corresponding to the true system is strictly decreasing, and this can be seen in Figure 8. In this figure, the solid line Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 367 represents the true cost function, while the dotted line represents the cost corresponding to model 3. It is interesting to observe that this last cost function is not decreasing, since the estimated state does not match exactly the true state. Note also that in the last period of time, the cost does not reach zero, as the new target is not inside the input feasible set. 2.5 r x 10 1.5i > 0.5 \ 50 time (min) 5 4.5 4 3.5 3 g 2.5 2 1.5 1 0.5 x 10 60 80 100 8 x10 7 ^ » 6 i \ 5 &-..: g 4 1 1 i 3 " 2 ] 1 n 100 150 time (min) Fig. 8. Cost function corresponding to the true system (solid line) and cost corresponding to model 3 (dotted line). Output j/max yi(°Q y 2 ( c) ■V3(°q 510 400 350 550 500 500 Table 3. New output zones for the FCC subsystem Next, we simulate a change in the output zones. The new bounds are given in Table 3. Corresponding to the new control zones, the input feasible set changes its dimension and shape significantly. In Figure 9, S* (@ x ) corresponds to the initial feasible set for the true model, and 3 U (6> a ), $^(<9 2 ) and 3 U (<9 3 ) represent the new input feasible sets for the three models considered in the robust controller. Since the input target is outside the input feasible set 3* = 3* (0 X ) fl &£ (@ 2 ) fl &£ (@ 3 ) , it is not possible to guide the system to a point in which the control cost is null at the end of the simulation time. When the output weight S y is as large as the input weight S U/ all the outputs are guided to their corresponding zones, while the inputs show a steady state offset with respect to the target u a des . The complete behavior of the outputs and inputs of the FCC subsystem, as well as the output set-points, can be seen in Figures 10 and 11, respectively when S =W 3 * diag(l 1 l) and S u = 10 3 * diag(l l) . The final stationary value of the input is u= [155 84], which represents the closest feasible input value to the target u a des . Finally, Figure 12 shows the control cost of 368 Robust Control, Theory and Applications the two simulated time periods. Observe that in the last period of time (from 51min to 100 min) the true cost function does not reach zero since the change in the operating point prevents the input and output constraints to be satisfied simultaneously. 100 90 80 70 60 40 final stationary u _l I I L 110 120 130 140 150 160 170 180 190 200 210 220 u1 Fig. 9. Input feasible sets for the FCC subsystem when a change in the output zones is introduced. 550 500 10 20 30 40 50 60 70 80 90 100 time (min) CO 600 400 70 80 90 100 10 20 30 40 50 6 time (min) Fig. 10. Controlled outputs and set points for the FCC subsystem with modified zones. Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 369 250 r 200 150 100 L 100 10 20 30 40 50 60 70 time (min) 90 100 CM 3 80 60 40 A i\ ; \r v^ ^^_ y^ \ \/^^ 10 20 30 40 50 60 70 80 90 100 time (min) Fig. 11. Manipulated inputs for the FCC subsystem with modified output zones. 10r 2.5 1.5 0.5 10 20 30 40 50 60 70 80 90 100 time (min) time (min) Fig. 12. Cost function for the FCC subsystem with modified zones. True cost function (solid line); Cost function of Model 3 (dotted line). 7. Conclusion In this chapter, a robust MPC previously presented in the literature was extended to the output zone control of time delayed system with input targets. To this end an extended 370 Robust Control, Theory and Applications model that incorporates additional states to account for the time delay is presented. The control structure assumes that model uncertainty can be represented as a discrete set of models (multi-model uncertainty). The proposed approach assures both, recursive feasibility and stability of the closed loop system. The main idea consists in using an extended set of variables in the control optimization problem, which includes the set point to each predicted output. This approach introduces additional degrees of freedom in the zone control problem. Stability is achieved by imposing non-increasing cost constraints that prevent the cost corresponding to the true plant to increase. The strategy was shown, by simulation, to have an adequate performance for a 2x3 subsystem of a typical industrial system. 8. References Badgwell T. A. (1997). Robust model predictive control of stable linear systems. International Journal of Control, 68, 797-818. Gonzalez A. H.; Odloak D.; Marchetti J. L. & Sotomayor O. (2006). IHMPC of a Heat- Exchanger Network. Chemical Engineering Research and Design, 84 (All), 1041-1050. Gonzalez A. H. & Odloak D. (2009). Stable MPC with zone control. Journal of Process Control, 19, 110-122. Gonzalez A. H.; Odloak D. & Marchetti J. L. (2009) Robust Model Predictive Control with zone control. IET Control Theory Appl., 3, (1), 121-135. Gonzalez A. H.; Odloak D. & Marchetti J. L. (2007). Extended robust predictive control of integrating systems. AIChE Journal, 53 1758-1769. Kassmann D. E.; Badgwell T. & Hawkings R. B. (2000). Robust target calculation for model predictive control. AIChE Journal, 45 (5), 1007-1023. Muske K.R. & Badgwell T. A. (2002). Disturbance modeling for offset free linear model predictive control. Journal of Process Control, 12, 617-632. Odloak D. (2004). Extended robust model predictive control. AIChE Journal, 50 (8) 1824-1836. Pannochia G. & Rawlings J. B. (2003). Disturbance models for offset-free model-predictive control. AIChE Journal, 49, 426-437. Qin S.J. & Badgwell T. A. (2003). A Survey of Industrial Model Predictive Control Technology, Control Engineering Practice, 11 (7), 733-764. Rawlings J. B. (2000). Tutorial overview of model predictive control. IEEE Control Systems Magazine, 38-52. Sotomayor O. A. Z. & Odloak D. (2005). Observer-based fault diagnosis in chemical plants. Chemical Engineering Journal, 112, 93-108. Zanin A. C; Gouvea M. T. & Odloak D. (2002). Integrating real time optimization into the model predictive controller of the FCC system. Contr. Eng. Pract., 10, 819-831. 16 Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method Shuxiang Guo Faculty of Mechanics, College of Science, Air Force Engineering University XV an 710051, P R China 1. Introduction Stability is of primary importance for any control systems. Stability of both linear and nonlinear uncertain systems has received a considerable attention in the past decades (see for example, Tanaka & Sugeno, 1992; Tanaka, Ikeda, & Wang, 1996; Feng, Cao, Kees, et al. 1997; Teixeira & Zak, 1999; Lee, Park, & Chen, 2001; Park, Kim, & Park, 2001; Chen, Liu, & Tong, 2006; Lam & Leung, 2007, and references therein). Fuzzy logical control (FLC) has proved to be a successful control approach for a great many complex nonlinear systems. Especially, the well-known Takagi-Sugeno (T-S) fuzzy model has become a convenient tool for dealing with complex nonlinear systems. T-S fuzzy model provides an effective representation of nonlinear systems with the aid of fuzzy sets, fuzzy rules and a set of local linear models. Once the fuzzy model is obtained, control design can be carried out via the so called parallel distributed compensation (PDC) approach, which employs multiple linear controllers corresponding to the locally linear plant models (Hong & Langari, 2000). It has been shown that the problems of controller synthesis of nonlinear systems described by the T-S fuzzy model can be reduced to convex problems involving linear matrix inequalities (LMIs) (Park, Kim, & Park, 2001). Many significant results on the stability and robust control of uncertain nonlinear systems using T-S fuzzy model have been reported (see for example, Hong, & Langari, 2000; Park, Kim, & Park, 2001; Xiu & Ren, 2005; Wu & Cai, 2006; Yoneyama, 2006; 2007), and considerable advances have been made. However, as stated in Guo (2010), many approaches for stability and robust control of uncertain systems are often characterized by conservatism when dealing with uncertainties. In practice, uncertainty exists in almost all engineering systems and is frequently a source of instability and deterioration of performance. So, uncertainty is one of the most important factors that have to be taken into account rationally in system analysis and synthesis. Moreover, it has been shown (Guo, 2010) that the increasing in conservatism in dealing with uncertainties by some traditional methods does not mean the increasing in reliability. So, it is significant to deal with uncertainties by means of reliability approach and to achieve a balance between reliability and performance/ control-cost in design of uncertain systems. In fact, traditional probabilistic reliability methods have ever been utilized as measures of stability, robustness, and active control effectiveness of uncertain structural systems by Spencer et al. (1992,1994); Breitung et al. (1998) and Venini & Mariani (1999) to develop 372 Robust Control, Theory and Applications robust control strategies which maximize the overall reliability of controlled structures. Robust control design of systems with parametric uncertainties have also been studied by Mengali and Pieracci (2000); Crespo and Kenny (2005). These works are meaningful in improving the reliability of uncertain controlled systems, and it has been shown that the use of reliability analysis may be rather helpful in evaluating the inherent uncertainties in system design. However, these works are within the probabilistic framework. In Guo (2007,2010), a non-probabilistic robust reliability method for dealing with bounded parametric uncertainties of linear controlled systems has been presented. The non- probabilistic procedure can be implemented more conveniently than probabilistic one whether in dealing with the uncertainty data or in controller design of uncertain systems, since complex computations are often associated with in controller design of uncertain systems. In this chapter, following the basic idea developed in Guo (2007, 2010), we focus on developing a robust reliability method for robust fuzzy controller design of uncertain nonlinear systems. 2. Problem statements and preliminary knowledge Consider a nonlinear uncertain system represented by the following T-S fuzzy model with parametric uncertainties: Plant Rule i: IF x x (t) is F n and ... andx n (t) is F in , THEN x(t) = A z (p)x(f ) + B z (p)u(f ), (i = 1, . . . , r ) U Where K is a fuzzy set, x(t) e R n is the state vector, u(t) e R m is the control input vector, r is the number of rules of the T-S fuzzy model. The system matrices A(p) and B(p) depend on the uncertain parameters p = { p x , p 2 , • • • , p v } . The defuzzified output of the fuzzy system can be represented by x(t ) = £ //,- (x(f))[ A, (p)x(t) + B, (p)u(t)] (2) In which JU { (*(*)) = CO, (*(*))/ ^ CO { (*(*)) , CD { (*(*)) = Y[ F ij ( X j V )) ( 3 ) / 1=1 j=\ Where F i j(Xj(t)) is the grade of membership of xAt) in the fuzzy set R- , a>i(x(t)) satisfies a>i (x(t)) > for all i ( i - 1,. . .,r ). Therefore, there exist the following relations ^■W0)^(i = l r), ^ Mi (x(t)) = \ (4) i=\ If the system (1) is local controllable, a fuzzy model of state feedback controller can be stated as follows: Control Rule i: IF x x (t) is F n and ... and x n (t) is F fn/ THEN u(t) = K i x(t) / (f = l,...,r) (5) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 373 Where Kj <=R mxn (f = l,...,r) are gain matrices to be determined. The final output of the fuzzy controller can be obtained by r u(t) = ^ Ml (x(t))K lX (t) (6) i=\ By substituting the control law (6) into (2), we obtain the closed-loop system as follows r r * (t) = Z Z Mi (x(t))M i (x(t))[ Ai (p) + Bi (p)K i ]x(t) (7) When the parameterized notation (Tuan, Apkarian, and Narikiyo 2001) is used, equations (6) and (7) can be rewritten respectively as «(0 = K(ju)x(t) (8) x(t) = {A(p,ju) + B(p,ju)K(ju))x(t) (9) Where M = (Mi(x(t)) 9 ... 9 Mx(t)))sn^l M GR^ (10) r r r K0i) = Y,M*(!Wi > A(p, J u) = Y J M l (x(t))A l (p) / B(p,ju) = Y,Vi(x(t))Bi(p) (11) i=\ f=l i=l Note that the uncertain parameters p = { pj , p 2 , • • • , p p } are involved in the expressions of (9) and (11). Following the basic idea developed by Guo (2007,2010), the uncertain-but-bounded parameters p = {pi,p 2 ,'-,p p } involved in the problem can be considered as interval variables and expressed in the following normalized form Pi=Pio+Pid#i (i = l,...,p) (12) where p iQ and p^ are respectively the nominal and deviational values of the uncertain parameter p { , Sj e [-1,1] is a standard interval variable. Furthermore, the system matrices are expressed in a corresponding form of that depend on the standard interval variables 5 = [Si,S 2 ,'",S p ] . Suppose that the stability of the control system can be reduced to solving a matrix inequality as follows M(5,P 1 ,P 2 ,...,P / )<0 (13) where, Pi,P 2 ,..,Pi are feasible matrices to be determined. The sign " < " indicates that the matrix is negative-definite. If the performance function (it may also be referred to as limit-state function) used for reliability analysis is defined in terms of the criterion (13) and represented by M = M(5,P 1 ,P 2 ,...,P/) , and the reliable domain in the space built by the standard variables 374 Robust Control, Theory and Applications 6 = [Si,S 2 ,--;S p ] is indicated by Q r = {5 : M(8, P { , P 2 , . . . , P z ) < 0} , then the robust reliability can be given as follows rj r = sup||5| oo :M(5,P 1 ,P 2 ,...,P z )<0}-l (14) 8eR? Where, \\d\\ denote the infinity norm of the vector 8 = [Si,S 2 , •••,$„]. Essentially, the robust reliability rj r defined by (14) represents the admissible maximum degree of expansion of the uncertain parameters in the infinity topology space built by the standard interval variables under the condition of that (13) is satisfied. If r/ r > holds, the system is reliable for all admissible uncertainties. The larger the value of r/ r , the more robust the system with respect to uncertainties and the system is more reliable for this reason. So it is referred to as robust reliability in the paper as that in Ben-Haim (1996) and Guo (2007,2010). The main objective of this chapter is to develop a method based on the robust reliability idea to deal with bounded parametric uncertainties of the system (1) and to obtain reliability- based robust fuzzy controller (6) for stabilizing the nonlinear system. Before deriving the main results, the following lemma is given to provide a basis. Lemma 1 (Guo, 2010). Given real matrices Y, E l ,E 2 ,---,E n , F l ,F 2 , ..., and F n with appropriate dimensions and Y = Y , then for any uncertain matrices A x = diag{Si j , • -' 9 S\ m } , A 2 =diag{S 2U '-,S 2m2 } ,•••, and A n =diag{S nl ,—,S nmn } satisfying \d^<a (i = l,...,n, j = 1,. ..,m n ), the following inequality holds for all admissible uncertainties Y + ^( £ , A F, + f; T 4 T E, T )<0 (15) 1=1 if and only if there exist n constant positive-definite diagonal matrices H { , H 2 , ..., and H n with appropriate dimensions such that n , , Y + ^yEiHiEf +a 2 F^H; l F i )<0 (16) i=\ 3. Methodology and main results 3.1 Basic theory The following commonly used Lyapunov function is considered V(x(t)) = x(t) T Px(t) (17) where P is a symmetric positive definite matrix. The time derivative of V(x(t)) is V(x(t)) = x(t) T Px(t) + x(tf Px(t) (18) Substituting (9) into (18), we can obtain y(x(0) = * T (0{(A(/^^ (19) So, V(x(t)) < is equivalent to (20) and further equivalent to (21) that are represented as follows Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 375 (A(p, M ) + B(p, M )K( M )) T P + P(A(p, M ) + B(p, M )K( M ))<0, M eI3 (20) {A(p, M )X + B(p, M )Y( M )) T +{A(p, M )X + B(p, M )Y(ju))<0 r ^ e n (21) In which, X = P~ , Y(ju) = K(ju)X possess the following form r r Y0/) = £/4(*(0)li =^Mx(t))K t X (i = l,...,r) (22) i'=i j=i Let Q ij (p,X,Y j ) = (A i (p)X + B i (p)Y j ) + (A i (p)X + B i (p)Y j ) T (i,j = \,...,r ) (23) then (21) can be written as r r Y^ViMjQij(p,X,*j)<0 (24) Some convex relaxations for (24) have been developed to make it tractable. Two type of relaxation are adopted here to illustrate the presented method. 3.1.1 A simple relaxation of (24) represented as follows is often used by authors (Lee, Park, & Chen 2001) Q ll (p,X,Y l )<0, Q ij (p,X,Y j ) + Q ji (p,X,Y i )<0 (l<i<j<r) (25) These expressions can be rewritten respectively as (A ; {p)X + B i (p)Y i ) + (A ; (p)X + B i (p)Y i ) T <0 (i = l,...,r) (26) (A,-(p)X + B,-(p)Y,-) + (A,(p)X + B,(p)Y ; ) T + (A ; (p)X + B^p)^) +(A ; (p)X + B y (p)Y,) T <0 (l<i<j<r) (27) Expressing all the uncertain parameters p = {p l ,p 2 ,--,p p } as the standard form of (12), furthermore, the system matrices are expressed as a corresponding form of that depend on the standard interval variables 8 = [Si,S 2 ,'~,S p ]. Without loss of generality, suppose that all the uncertain matrices A z (p) and B z (p) can be expressed as V 1 A i (p) = A io +Y, A ij S ij< B i(P) = B io + Ys BikSik ( i = l > — r ) ( 28 ) In which, A i0 , B i0 , A» , and B ik are known real constant matrices determined by the nominal and deviational values of the basic variables. To reduce the conservatism caused by dealing with uncertainties as far as possible, representing all the matrices A» and B ik as the form of the vector products as follows 376 Robust Control, Theory and Applications Aq=VinV$ 2 , B lk =U, n ul 2 («=l,...,r, j=\...,V. k = l,...,q) In which, V#i / ^72 ' ^zfci / anc ^ ^ikl are a ^ column vectors. Denoting (29) V,i=[v,n V, 2 V . V,2=[V, 12 v, 2 >2 u,i=[u,ii u,2i - u,,i]; u, 2 = [u,i2 u,22 - u„ 2 ] r , A,-i = <%{$! ,• • • , <5j p }; A, 2 = rfkg{ J n ,• • • , S iq }; (i = l,...,r) (30) where, the first four matrices are constructed by the column vectors involved in (29). Then, the expressions in (28) can be further written as A i (p) = A i0 +V n A n V i2 , B i (p) = B i0 +U il A i2 U i2 (i=l r) (31) Substituting (31) into equations (26) and (27), we can obtain (A, X + B, Y,) + (A, X + B, Y,) T + (V fl A fl V„X) + (V fl A fl V f2 X) r +(U,A2U, 2 Y,) + (U, 1 A, 2 U, 2 Y,) T <0 (i = l r) (A, X + B,- Y, ) + (A, X + B, Y, ) T + (V n A a V i2 X) + ( V fl A n V, 2 X) T + (UaA,-2U,. 2 Y i ) + (U,. 1 A i2 U i2 Y i ) T + +(A ;0 X + B ;0 Y,) + (A y0 X + B ;0 Y,) T + (V ;1 A ;1 V y2 X) + (V ;1 A ;1 V ;2 X) T + (U ; iA ; . 2 U ;2 Y,) + (U ;1 A ; . 2 U ;2 Y,) r < (1 < i < j < r) In terms of Lemma 1, the matrix inequality (32) holds for all admissible uncertainties if and only if there exist diagonal positive-definite matrices E zl and E i2 with appropriate dimensions such that (32) (33) (A, X + B, Y,) + (A, X + B, Y,) T + V (1 E,X + a 2 (V f2 X) T E;?(V f2 X) + +U il E i2 U[ 1 +« 2 (U <2 Y,) T E- 2 1 (U, 2 Y,)<0 (i = l r) (34) Similarly, (33) holds for all admissible uncertainties if and only if there exist constant diagonal positive-definite matrices H«i , H^ 2 , Hm , and Hy 4 such that (A,- X + B, Y ; ) + (A,- X + B, Y y ) T + V (1 H p V^ + V /a Hg 2 v£ + +U, 1 H <;3 Uf 1 + U ;1 H §4 U)i + (A y0 X + B ; . Y,) + (A ;0 X + B ;0 Y,) T + a 2 (V, 2 X) r H^(V, 2 X) + a 2 (V ;2 X) r H^(V ;2 X) + +a 2 (U, 2 Y 7 ) r H^ 3 (U, 2 Y 7 ) + « 2 (U y2 Y,) T H^ 4 (U y2 Y,) < (1 < i < j < r) Applying the well-known Schur complement, (34) and (35) can be written respectively as (35) aXVf 2 (aU i2 Yj) T aV l2 X -E n aU i2 Y <0 (i = l,...,r) (36) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 377 Ta aV l2 X - Hij X * aV j2 X -H aU i2 Yj aUj 2 Yi ij2 H ifl - Hij 4 <0 ( 1 < z < ; < r J (37) In which, S, = (A, X + B, Y,.) + (A,. X + B,. Y,.) T + V a E a vS + U n E i2 ul, r, y = (A, X + B, Y ; .) + (A, X + B, Y ; .) T + {A^X + B^ + iA^X + B^f + V a H^yl + v n n ij2 vl + UnHpuZ + u n H ijA u T n . denotes the transposed matrices in the symmetric positions. Consequently, the following theorem can be obtained. Theorem 3.1. For the dynamic system (2) with the uncertain matrices represented by (31) and \<5 m | < a ( m = 1,. . .,/? ), it is asymptotically stabilizable with the state feedback controller (6) if there exist a symmetric positive-definite matrix X , matrices Y z , and constant diagonal positive-definite matrices E zl , E i2 , H^ , H i; - 2 > H« 3 , and H z ; 4 ( 1 < i < j < r ) such that the LMIs represented by (36) and (37) hold for all admissible uncertainties. If the feasible matrices X and Y z are found out, then the feedback gain matrices deriving the fuzzy controller (6) can be obtained by K i =Y i X' 1 (z = l,...,r) (38) It should be stated that the condition of (25) is restrictive in practice. It is adopted yet here is merely to show the proposed reliability method and for comparison. 3.1.2 Some improved relaxation for (24) have also been proposed in literatures. A relaxation provided by Tuan, Apkarian, and Narikiyo (2001) is as follows Q„(P,X,Y ; )<0 (i=l r) Qu(p,X,Y i ) + ^-(Q ll (p,X,Y J ) + Q JI (p,X,Y l ))<0(lZi*iZr) (39) (40) The expression (39) is the same completely with the first expression of (25). So, only (40) is investigated further. It can be rewritten as (A,.(p)X + B ; (p)Y,) + (A,.(p)X + B,(p)Y,.) r + ^{(A 1 .(p)X + B,.(p)Y ; ) + (A 1 .(p)X + B,(p)Y ; ) T + +(A ; (p)X + B ; (p)Y,.) + (A ; (p)X + B ; (p)Y ; ) r }<0 (l<i*;<r) On substituting the expression (31) into (41), we obtain (41) 378 Robust Control, Theory and Applications {(A i0 X + B i0 Y f ) + (A f0 X + B f0 Y f ) T } + ^{(A i0 X + B z0 Y ; ) + (A i0 X + B z0 Y ; ) T } + + ^i{(A ;0 X + B ;0 Y Z ) + (A ;0 X + B ;0 Y Z ) T } + ^{(V fl A fl V i2 X) + (V fl A fl V i2 X) T } + +(U ll A f2 U i2 Y i ) + (U zl A z2 U z2 Y z ) T + (^U zl A z2 U z2 Y ; ) + (^U zl A z2 U z2 Y ; ) T {(V ;1 A ;1 V ;2 X) + (V ;1 A ;1 V ;2 X) T + (U ;1 A ; . 2 U ; . 2 Y f ) + (U ;1 A ;2 U ;2 Y Z ) T } < (1 < i * j < r) (42) r-1 2 In terms of Lemma 1, the matrix inequality (42) hold for all admissible uncertainties if and only if there exist constant diagonal positive-definite matrices F n , F i2 , F i3 , H^ , and H i; - 2 such that {(A z0 X + B i0 Y f ) + (A i0 X + B i0 Y f ) T } + ^{(A i0 X + B i0 Y ; .) + (A i0 X + B i0 Y/} r-1 v 2/tr V\Tt7-1/ 2 {(A ; . X + B j0 X) + (A ; . X + B ; . Y,) r j + [ — V (1 |F n | ^ V, , | -t a\ V, 2 X)< F ;1 ' (V, 2 X) +U, 1 F, 2 Uf 1 +« 2 (U, 2 Y,) r F- 2 1 (U, 2 Y,)} + ^U, 1 jH, yl ^U, 1 J + « 2 (U, 2 Y 7 ) r H^(U, 2 Y y ) (43) ^V ;1 ]F ; ,[l^V ;1 ]\[l^U ;1 ]H, 2 [l^U ;1 ]\« 2 (V ;2 X) r F-(V ;2 X) + « 2 (U ; . 2 Y,) r H^ 2 (U ; . 2 Y,)} < (1 < i * j < r) Applying the Schur complement, (43) is equivalent to w V * * aV i2 X -F,i * aU l2 Y t -F i2 aV j2 X -F j3 * aU i2 Y j -H in aU j2 Y { H ,/2 <0 (\<i*j<r) (44) r-1 In which, W t] = {(A, X + B, Y,) + (A, X + B, Y,) T } +— {(A, X + B, Y ; ) + (A, X + B, Y ; ) T } h^{(A ;0 X + B y0 Y,) + (A ;0 X + B y0 Y,) T } + f ^V a jp a ^V ;l j + U fl F f2 u£ + 2 r-1 V,! R ;i r ;3 r-1 r-1, 2" V H + ^ Ul1 % 2 r-1 r-1, 2~ U a | + |— U ;1 H, y2 ^ . • This can be summarized as follows. Theorem 3.2. For the dynamic system (2) with the uncertain matrices represented by (31) and \S m \ < a (m = l,...,p ), it is asymptotically stabilizable with the state feedback controller (6) if there exist a symmetric positive-definite matrix X , matrices Y f , and constant diagonal Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 379 positive-definite matrices E a , E i2 , F n , F i2 , F i3 , H^ , and H i; - 2 (l<i^ j <r) such that the LMIs (36) and (44) hold for all admissible uncertainties. If the feasible matrices X and Y { are found out, the feedback gain matrices deriving the fuzzy controller (6) can then be given by (38). 3.2 Robust reliability based stabilization In terms of Theorem 3.1, the closed-loop fuzzy system (7) is stable if all the matrix inequalities (36) and (37) hold for all admissible uncertainties. So, the performance functions used for calculation of reliability of that the uncertain system to be stable can be taken as M l (a,X,Y n E n ,E l2 ) = M^a 9 XX,Yj,Hyl,Hij2,Hy3,Hij4) = 1—1 i aXV] 2 (aU l2 Y t ) T aV i2 X ~E n aU i2 Y t ~E i2 r -j * * * aV i2 X -H-iji * * aV j2 X - H * n ij2 aU il Y j -H aU ]2 Y t (i = l,...,r) (45) //3 -H ijA (1 < i < j < r) (46) in which, the expressions of E { and T f are in the same form respectively as that in (36) and (37). Therefore, the robust reliability of the uncertain nonlinear system in the sense of stability can be expressed as /7 r = sup {tf:M z KX,X,E zl ,E^^ (47) aeR + where, R + denotes the set of all positive real numbers. The robust reliability of that the uncertain closed-loop system (7) is stable may be obtained by solving the following optimization problem Maximize a Subject to M,(«,X,Y,,E, 1 ,E, 2 )<0 M, j (a ; X,Y,Y i ,H, n ,H,p,H, ]3 ,H,- 4 )<0 E n > 0, E, 2 > X > 0, H.j > 0, H,- 2 > 0, H, 3 > 0, H,- 4 > (1 < i < j < r) (48) From the viewpoint of robust stabilizing controller design, if inequalities (36) and (37) hold for all admissible uncertainties, then there exists a fuzzy controller (6) such that the closed- loop system (7) to be asymptotically stable. Therefore, the performance functions used for reliability-based design of control to stabilize the uncertain system (2) can also be taken as that of (45) and (46). So, a possible stabilizing controller satisfying the robust reliability requirement can be given by a feasible solution of the following matrix inequalities M,(a ,X,Y,,E, 1 ,E, 2 )<0, M {j {a ,X,Y J ,Y y ,Hp,H f/2 ,H f/3 ,H ( ,. 4 )<0; E, 1 >0E, 2 >0, X>0 / Hp>0,H, 2 >0 / H !/3 >0,H ;4 >0 (l<i<j<r) a =-q a +\ (49) 380 Robust Control, Theory and Applications In which, M { (•) and M,y (•) are functions of some matrices and represented by (45) and (46) respectively. r/ cr is the allowable robust reliability. If the control cost is taken into account, the robust reliability based design optimization of stabilization controller can be carried out by solving the following optimization problem Minimize Trace(N); Subject to M { (a ,X,%,E n ,E i2 ) < M zy (a ,X, Y f/ Y^H^H^H^,] E fl >0, E f2 >0, H tjl >0, H zy2 >0, H zy3 >0, % >0, (l<z<;<r) n r (50) I X >0, X>0,N>0, a =?] cr +l In which, the introduced additional matrix N is symmetric positive-definite and with the same dimension as X . When the feasible matrices X and Y f are found out, the optimal fuzzy controller could be obtained by using (6) together with (38). If Theorem 3.2 is used, the expression of M {] ,(■) corresponding to (46) becomes M ii (a,X,Y i ,Y j ,F a ,F i2 ,F a ,H ijl ,H ii 2)- V„ aV i2 X ~Fn X- * aU i2 Y { -Fa * aV j2 X -F aU i2 Y j a\l ]2 Y x o H ijl (l<i*j<r) (51) where, W^ is the same with that in (44). Correspondingly, (47) and (48) become respectively as follows. ^ = sup{a:M,(a,X / Y, / E, 1/ E, 2 )<0 / M^(a,X,Y,,Y ; ,F, 1 ,F, 2/ F, 3/ H, yl ,H y . 2 )<0, aeR + E fl >0, E i2 >0, X>0,F zl >0,F z2 >0,F z3 >0,H z;/1 >0,Hy 2 >0, l<i*j<r}-l (52) Maximize a Subject to M z («,X,Y z ,E zl ,E z2 ) < M zy (a,X,Y z ,Y 7 ,F zl ,F z2 ,F z3 ,H zyi ,H z;2 ) < E zl > 0,E f2 > 0,X > 0,F fl > 0,F z2 > 0,F z3 > 0,H z;/1 > 0,H z)2 > 0, (l<i*j< r) Similarly, (50) becomes (53) Minimize Trace(N) Subject to M f (a*,X, Y z , E zl , E z2 ) < 0, M zy (a\X,Y z ,Y ; ,F zl ,F z2 ,F z3 ,H zyi ,H f;2 ) < E zl >0,E z2 >0,F zl >0,F z2 >0,F z3 >0,H zyi >0,H zy2 >0, (l<i±j<r) n r I X >0, X>0,N>0, a =?] cr +l (54) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 381 3.3 A special case Now, we consider a special case in which the matrices of (30) is expressed as V a =V„ V i2 = V 2 , A n =A, U a =U i2 =0 (» = 1 r) (55) This means that the matrices A, (p) in all the rules have the same uncertainty structure and the matrices B, (p) become certain. In this case, (31) can be written as A i (p) = A l0 +V l AV 2 , B,(p) = B t0 (» = l,...,r) (56) and the expressions involved in Theorem 3.1 can be simplified further. This is summarized in the following. Theorem 3.3. For the dynamic system (2) with the matrices represented by (56) and \S m \ < a (m = l,...,p), it is asymptotically stabilizable with the state feedback controller (6) if there exist a symmetric positive-definite matrix X , matrices Y t , and constant diagonal positive- definite matrices E { and H^ (\<i< j <r) with appropriate dimensions such that the following LMIs hold for all admissible uncertainties dXVi aV 2 X <0, aV 2 X aXV 2 L < ( 1 < i < j < r ) (57) In which, E t = (A l0 X + B l0 Y l ) + (A l0 X + B l0 Y l ) T + V l E ll V l T f I {] = (A i0 X + B i0 Y j ) + ( A i0 X + B i0 Y j ) T + (A ;0 X + B j0 Y { ) + ( A j0 X + B j0 Y t ) T + (2V 1 )H- ; (2y i ) T . If the feasible matrices X and Y { are found out, the feedback gain matrices deriving the fuzzy controller (6) can then be given by (38). Proof. In the case, inequalities (32) and (33) become, respectively, (A z0 X + B 20 Y 2 ) + (A 20 X + B 20 Y 2 ) r +(V 1 Ay 2 X) + (y i Ay 2 X) T <0 (i = l r) (58) (A z0 X + B z0 Y ; ) + (A z0 X + B z0 Y j ) T +2(V 1 A V 2 X) + 2(V a A V 2 X) T +(A ;0 X + B ;0 Y Z ) + (A ;0 X + B ;0 Y Z ) T < (1 < i < j < r) (59) In terms of Lemma 1, (58) holds for all admissible uncertainties if and only if there exist diagonal positive-definite matrices E { ( i = 1, . . .,r ) with appropriate dimensions such that (A i0 X + B, Y,) + (A, X + B i0 X) T + V X E { V? + a 2 (V 2 X) T E; l (V 2 X)<0 (i = l,...,r) (60) Similarly, (59) holds for all admissible uncertainties if and only if there exist constant diagonal positive-definite matrices EL- such that (A, X + B,. Y ; .) + (A, X + B, Y ; .) T + (A j0 X + B j0 Y ; ) + +(A ;0 X + B ;0 Y,) T + (2V 1 )H # (2V 1 ) T + « 2 (V 2 X) T H^(V 2 X) < (1 < i < j < r) (61) Applying Schur complement, (57) can be obtained. So, the theorem holds. By Theorem 3.3, the performance functions used for reliability calculation can be taken as 382 Robust Control, Theory and Applications M i (a / X / Y f/ E i ) = aXVi aV 7 X , (i = l,...,r) M f/ (a / X / Y i/ Y // H i/ ) = aXVj aV 2 X -H f; (62) (1 < f < j < r) Accordingly, a possible stabilizing controller satisfying robust reliability requirement can be obtained by a feasible solution of the following matrix inequalities M i (a,X,Y i ,E i )<0,M ij (a,X,\ i ,Y j ,H ij )<0, X>0, E z >0; H i; >0 (l<i<j<r) a = r/ cr + 1 (63) The optimum stabilizing controller based on the robust reliability and control cost can be obtained by solving the following optimization problem Minimize Trace(N) Subject to M { (a ,X, Y f/ E f ) < 0, M {j (a ,X,\, Y ; -,H i; .) < 0; E f > 0,Hy > 0, (1 < z < ; < r) (64) N r I X >0, X>0,N>0, a =?] cr +l Similarly, the expressions involved in Theorem 3.2 can also be simplified and the corresponding result is summarized in the following. Theorem 3.4. For the dynamic system (2) with the matrices represented by (56) and \S m | < a (m = l,...,p), it is asymptotically stabilizable with the state feedback controller (6) if there exist a symmetric positive-definite matrix X , matrices Y { , and constant diagonal positive- definite matrices E { and H^ ( 1 < z ^ j <r) with appropriate dimensions such that the following LMIs hold for all admissible uncertainties aXVj aV 2 X <0, w. (aV 2 Xf (aV 2 X) H V1 < ( 1 < i * j < r ; (65) In which, S t = (A l0 X + B l0 Y l ) + (A l0 X + B l0 Y l ) T + V^V* , W {j = (rV^irV.f+iiA^X + B^) + (A l0 X + B l0 Y l ) T ) + L^{(A l0 X + B l0 Y j ) + (A l0 X + B l0 Y j ) T + (A j0 X + B j0 Y l ) + (A j0 X + B j0 Y l ) T }. If the feasible matrices X and Y; are found out, the feedback gain matrices deriving the fuzzy controller (6) can then be obtained by (38). Proof. (42) can be rewritten as {(A, X + B, Y,) + (A,- X + B, Y,) T } + ^-{(A, X + B, Y ; ) + (A, X + B, Y ; ) T + (A ;0 X + B ;0 Y,) + (A ;0 X + B ;0 Y,) T j + r {(V X A V 2 X) + (V X A V 2 X) T j < (1 < i * ; < r ) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 383 In terms of Lemma 1, (66) holds for all admissible uncertainties if and only if there exist diagonal positive-definite matrices EL- such that r-1 {(A i0 X + B i0 Y f ) + (A i0 X + B i0 Y f ) T } + ^-{(A f0 X + B z0 Y ; ) + (A f0 X + B z0 Y ; ) T + (A ;0 X + B ;0 Y Z ) + (A ;0 X + B ;0 Y z ) T } + (rV 1 )H zy (rV 1 ) T + a 2 (V 2 X) T H^ 1 (V 2 X)<0 l<i*j<r (67) Applying Schur complement, (67) is equivalent to the second expression of (65). So, the theorem holds. By Theorem 3.4, the performance functions used for reliability calculation can be taken as M z (a,X,Y z ,E z ; (aV 2 xf («v 2 x) , (i = l,...,r) (68) M iy (a / X / Y i/ Y ; . / H^.) = W* (aV 2 X) J (aV 2 X) H„ , (l<i*j<r) So, design of the optimal controller based on the robust reliability and control cost could be carried out by solving the following optimization problem Minimize Trace(N) Subject to M 2 (a , X,Y z ,E z )<0, M^c^X^Y^H^) <0, E { >0,H i; - >0 (l<i*j<r) (69) N I I X >0, X>0,N>0 (a =rj cr +l) 4. Numerical examples Example 1. Consider a simple uncertain nonlinear mass-spring-damper mechanical system with the following dynamic equation (Tanaka, Ikeda, & Wang 1996) x(t) + x(t) + c(t)x(t) = (1 + 0. 1 3x 3 (t))u(t) Where c(t) is the uncertain term satisfying c(t) e [0.5,1.81] . Assume that x(t) e [-1.5,1.5] , x(t) e [-1.5,1.5] . Using the following fuzzy sets FJx(t)) = 0.5 + ?-&, F 2 (x(t)) = 0.5-^-0- 1V v " 6.75 2V V " 6.75 The uncertain nonlinear system can be represented by the following fuzzy model Plant Rule 1: IF x(t) is about F { , THEN x(t) = A x x(t) + B x ii(t) Plant Rule 2: IF x(t) is about F 2 ,THEN x(t) = A 2 x(t) + B 2 u(t) Where x(t) = m x(t) , A 1 - A 2 ~-l -c 1 ,B 1 = "1.43875" ,B 2 = "0.56125" 384 Robust Control, Theory and Applications Expressing the uncertain parameter c as the normalized form, c = 1.155 + 0.655£, furthermore, the system matrices are expressed as A^A^+V.AV,, A 2 = A 20 +V { AV 2 , B { =B l0 , B 2 =B 20 . In which ^10 ~~ ^-20 - -1 -1.155 1 >Vi = V 2 =[0 -0.655], A = S . By solving the optimization problem of (69) with a = 1 and a = 3 respectively, the gain matrices are obtained as follows K 1= [-0.0567 -0.1446], K 2 = [-0.0730 -0.1768] (a* = 1 ); #!= [-0.3645 -1.0570], K 2 = [-0.9191 -2.4978] (a =?>). When the initial value of the state is taken as x(0) = [- 1 -1.3] , the simulation results of the controlled system with the uncertain parameter generated randomly within the allowable range c(t) e [0.5,1.81] are shown in Fig. 1. Time (sec) Time (sec) Fig. 1. Simulation of state trajectories of the controlled system (The uncertain parameter c is generated randomly within [0.5, 1.81]) Example 2. Consider the problem of stabilizing the chaotic Lorenz system with parametric uncertainties as follows (Lee, Park, & Chen, 2001) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 385 pi (')] x 2 (t) = |*30)J -crx 1 (f) + ax 2 (t) r Xl (t)-x 2 (t)- Xl (t)x 3 (t) Xl (t)x 2 (t)-bx 3 (t) For the purpose of comparison, the T-S fuzzy model of the chaotic Lorenz system is constructed as Plant Rule 1 : IF x x {t) is about M 1 THEN x(t) = A a x(f) + B 1 u(*) Plant Rule 2 : IF x x (t) is about M 2 THEN x(t) = A 2 x(t) + B 2 u(t) Where A 1 = r -1 M 1 -M 1 , A 2 = -a a r -1 -M 2 M 2 -b The input matrices B x and B 2 , and the membership functions are respectively The nominal values of (cr,r,b) are (10, 28, 8/3), and choosing [M 1? M 2 ] = [-20,30] . All system parameters are uncertain-but-bounded within 30% of their nominal values. The gain matrices for deriving the stabilizing controller (6) given by Lee, Park, and Chen (2001) are K lL = [-295.7653 -137.2603 -8.0866], K 2L = [-443.0647 -204.8089 12.6930]. (1) Reliability-based feasible solutions In order to apply the presented method, all the uncertain parameters (<j, r, b) are expressed as the following normalized form a = \0 + 3S { , r = 28 + 8.4£ 2 , & = 8/3 + 0.8<? 3 . Furthermore, the system matrices can be expressed as - V X AV 2 , A 2 = A 20 + V X AV 2 , B x = B 10 , B 2 = B 20 . A 1~ Ao In which "-10 10 A 10 = 28 -1 20 -20 -8/3 -10 10 28 -1 30 "1 0" -30 ,v 1 = 10 /V 2 = ]/3_ 1 -3 3 8.4 -0.8 A = diag{S 1 ,S 2 ,S 3 }, B 10 =B 20 =[1 of By solving the matrix inequalities corresponding to (63) with a* =1 , the gain matrices are found to be K a = [-84.2940 -23.7152 -2.4514], K 2 =[-84.4669 -23.6170 3.8484] 386 Robust Control, Theory and Applications The common positive definite matrix X and other feasible matrices obtained are as follows X = 0.0461 -0.0845 0.0009 -0.0845 0.6627 -0.0027 0.0009 -0.0027 0.6685 , E 1 =diag{3.4260,2.3716,1.8813} E 2 =diag{3. 4483,2.2766,1. 9972}, H = diag{l. 653 5,1. 9734,1 .33 18}. Again, by solving the matrix inequalities corresponding to (63) with a* = 2, which means that the allowable variation of all the uncertain parameters are within 60% of their nominal values, we obtain 1^= [-123.6352 -42.4674 -4.4747], K 2 = [-125.9081 -42.8129 6.8254], X E 2 = d\ 1.0410 -1.7704 0.0115 -1.7704 7.7159 -0.0360 0.0115 -0.0360 7.7771 !101.9235,42.7451,24.7517}, H = , E { =^{98.7271,44.0157,22.7070}, 1.8833,31.0026,13. 8173}. Clearly, the control inputs of the controllers obtained in the paper in the two cases are all lower than that of Lee, Park, and Chen (2001). 20 -20 v\f\f\ *,(t) 10 20 -20 WvAAf xAt) 10 60 40 20 x,(t) 60 40 20 10 x,(t) Time (sec) Time (sec) 10 Fig. 2. State trajectories of the controlled nominal chaotic Lorenz system (On the left- and right-hand sides are results respectively of the controller of Lee, Park, and Chen (2001) and of the controller obtained in this paper) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 387 (2) Robust reliability based design of optimal controller Firstly, if Theorem 3.3 is used, by solving a optimization problem corresponding to (64) with a = 1 , the gain matrices as follows for deriving the controller are obtained K lG = [-20.8512 -13.5211 -3.2536], K 2G = [-21.2143 -13.1299 4.3799]. The norm of the gain matrices are respectively ]K 1G || = 25.0635 and |JC 2G || = 25.3303 . So, there exist relations K 1L =326.1639 = 13.0135 K 1G \\Kor =- ;.2767 = 19.2764 K 2G To examine the effect of the controllers, the initial values of the states of the Lorenz system are taken as x(0) = [lO -10 -10] , the control input is activated at £=3.89s, all as that of Lee, Park, and Chen (2001), the simulated state trajectories of the controlled Lorenz system without uncertainty are shown in Fig. 2. In which, on the left- and right-hand sides are results of the controller of Lee, Park, and Chen (2001) and of the controller obtained in this paper respectively. Simulations of the corresponding control inputs are shown in Fig. 3, in which, the dash-dot line and the solid line represent respectively the input of the controller of Lee, Park, and Chen (2001) and of the controller in the paper. 6000 4000 2000 4.5 5 Time (sec) Fig. 3. Control input of the two controllers (dash-dot line and solid line represent respectively the result of Lee, Park, and Chen (2001) and the result of the paper) The simulated state trajectories and phase trajectory of the controlled Lorenz system are shown respectively in Figs. 4 and 5, in which, all the uncertain parameters are generated randomly within the allowable ranges. 388 Robust Control, Theory and Applications 20 -20 |jU| *° ■ NfjF 20 -20 I^Aufi * 2(t) ' w|f 5 Time (sec) Time (sec) Fig. 4. Ten-times simulated state trajectories of the controlled chaotic Lorenz system with parametric uncertainties (all uncertain parameters are generated randomly within the allowable ranges, and on the left- and right-hand sides are respectively the results of controllers in Lee, Park, and Chen (2001) and in the paper) 40 50 Fig. 5. Ten-times simulated phase trajectories of the parametric uncertain Lorenz system controlled by the presented method (all parameters are generated randomly within their allowable ranges) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 389 It can be seen that the controller obtained by the presented method is effective, and the control effect has no evident difference with that of the controller in Lee, Park, and Chen (2001), but the control input of it is much lower. This shows that the presented method is much less conservative. Taking a = 3 , which means that the allowable variation of all the uncertain parameters are within 90% of their nominal values, by applying Theorem 3.3 and solving a corresponding optimization problem of (64) with a = 3 , the gain matrices for deriving the fuzzy controller obtained by the presented method become K 1G = [-54.0211 -32.5959 -6.5886], K 2G =[-50.0340 -30.6071 10.4215]. Obviously, the input of the controller in this case is also much lower than that of the controller obtained by Lee, Park, and Chen (2001). Secondly, when Theorem 3.4 is used, by solving two optimization problems corresponding to (69) with a = 1 and a = 3 respectively, the gain matrices for deriving the controller are found to be K 1G = [-20.8198 -13.5543 -3.2560], K 2G =[-21.1621 -13.1451 4.3928] (a =1). K 1G = [-54.0517 -32.6216 -6.6078], K 2G =[-50.0276 -30.6484 10.4362] (a = 3) Note that the results based on Theorem 3.4 are in agreement, approximately, with those based on Theorem 3.3. 5. Conclusion In this chapter, stability of parametric uncertain nonlinear systems was studied from a new point of view. A robust reliability procedure was presented to deal with bounded parametric uncertainties involved in fuzzy control of nonlinear systems. In the method, the T-S fuzzy model was adopted for fuzzy modeling of nonlinear systems, and the parallel- distributed compensation (PDC) approach was used to control design. The stabilizing controller design of uncertain nonlinear systems were carried out by solving a set of linear matrix inequalities (LMIs) subjected to robust reliability for feasible solutions, or by solving a robust reliability based optimization problem to obtain optimal controller. In the optimal controller design, both the robustness with respect to uncertainties and control cost can be taken into account simultaneously. Formulations used for analysis and synthesis are within the framework of LMIs and thus can be carried out conveniently. It is demonstrated, via numerical simulations of control of a simple mechanical system and of the chaotic Lorenz system, that the presented method is much less conservative and is effective and feasible. Moreover, the bounds of uncertain parameters are not required strictly in the presented method. So, it is applicable for both the cases that the bounds of uncertain parameters are known and unknown. 6. References Ben-Haim, Y. (1996). Robust Reliability in the Mechanical Sciences, Berlin: Spring- Verlag Breitung, K.; Casciati, F. & Faravelli, L. (1998). Reliability based stability analysis for actively controlled structures. Engineering Structures, Vol. 20, No. 3, 211-215 390 Robust Control, Theory and Applications Chen, B.; Liu, X. & Tong, S. (2006). Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay. Fuzzy Sets and Systems, Vol. 157, 2224-2240 Crespo, L. G. & Kenny, S. P. (2005). Reliability-based control design for uncertain systems. Journal of Guidance, Control, and Dynamics, Vol. 28, No. 4, 649-658 Feng, G.; Cao, S. G.; Kees, N. W. & Chak, C. K. (1997). Design of fuzzy control systems with guaranteed stability. Fuzzy Sets and Systems, Vol. 85, 1-10 Guo, S. X. (2010). Robust reliability as a measure of stability of controlled dynamic systems with bounded uncertain parameters. Journal of Vibration and Control, Vol. 16, No. 9, 1351-1368 Guo, S. X. (2007). Robust reliability method for optimal guaranteed cost control of parametric uncertain systems. Proceedings of IEEE International Conference on Control and Automation, 2925-2928, Guangzhou, China Hong, S. K. & Langari, R. (2000). An LMI-based Hoo fuzzy control system design with TS framework. Information Sciences, Vol. 123, 163-179 Lam, H. K. & Leung, F. H. F. (2007). Fuzzy controller with stability and performance rules for nonlinear systems. Fuzzy Sets and Systems, Vol. 158, 147-163 Lee, H. J.; Park, J. B. & Chen, G. (2001). Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Transactions on Fuzzy Systems, Vol. 9, 369-379 Park, J.; Kim, J. & Park, D. (2001). LMI-based design of stabilizing fuzzy controllers for nonlinear systems described by Takagi-Sugeno fuzzy model. Fuzzy Sets and Systems, Vol. 122, 73-82 Spencer, B. F.; Sain, M. K.; Kantor, J. C. & Montemagno, C. (1992). Probabilistic stability measures for controlled structures subject to real parameter uncertainties. Smart Materials and Structures, Vol. 1, 294-305 Spencer, B. F.; Sain, M. K.; Won C. H.; et al. (1994). Reliability-based measures of structural control robustness. Structural Safety, Vol. 15, No. 2, 111-129 Tanaka, K.; Ikeda, T. & Wang, H. O. (1996). Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, FL control theory, and linear matrix inequalities. IEEE Transactions on Fuzzy Systems, Vol. 4, No. 1, 1-13 Tanaka, K. & Sugeno, M. (1992). Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems, Vol. 45, 135-156 Teixeira, M. C. M. & Zak, S. H. (1999). Stabilizing controller design for uncertain nonlinear systems using fuzzy models. IEEE Transactions on Fuzzy Systems, Vol. 7, 133-142 Tuan, H. D. & Apkarian, P. (1999). Relaxation of parameterized LMIs with control applications. International Journal of Nonlinear Robust Control, Vol. 9, 59-84 Tuan, H. D.; Apkarian, P. & Narikiyo, T. (2001). Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Transactions on Fuzzy Systems, Vol. 9,324-333 Venini, P. & Mariani, C. (1999). Reliability as a measure of active control effectiveness. Computers and Structures, Vol. 73, 465-473 Wu, H. N. & Cai, K. Y. (2006). H2 guaranteed cost fuzzy control design for discrete-time nonlinear systems with parameter uncertainty. Automatica, Vol. 42, 1183-1188 Xiu, Z. H. & Ren, G. (2005). Stability analysis and systematic design of Takagi-Sugeno fuzzy control systems. Fuzzy Sets and Systems, Vol. 151, 119-138 Yoneyama, J. (2006). Robust Hoo control analysis and synthesis for Takagi-Sugeno general uncertain fuzzy systems. Fuzzy Sets and Systems, Vol. 157, 2205-2223 Yoneyama, J. (2007). Robust stability and stabilization for uncertain Takagi-Sugeno fuzzy time-delay systems. Fuzzy Sets and Systems, Vol. 158, 115-134 17 A Frequency Domain Quantitative Technique for Robust Control System Design Jose Luis Guzman 1 , Jose Carlos Moreno 2 , Manuel Berenguel 3 , Francisco Rodriguez 4 , Julian Sanchez-Hermosilla 5 1/2/3/4l Departamento de Lenguajes y Computation; 5 Departamento de Ingenieria Rural, University of Almeria Spain 1. Introduction Most control techniques require the use of a plant model during the design phase in order to tune the controller parameters. The mathematical models are an approximation of real systems and contain imperfections by several reasons: use of low-order descriptions, unmodelled dynamics, obtaining linear models for a specific operating point (working with poor performance outside of this working point), etc. Therefore, control techniques that work without taking into account these modelling errors, use a fixed-structure model and known parameters (nominal model ) supposing that the model exactly represents the real process, and the imperfections will be removed by means of feedback. However, there exist other control methods called robust control techniques which use these imperfections implicity during the design phase. In the robust control field such imperfections are called uncertainties, and instead of working only with one model (nominal model), a family of models is used forming the nominal model + uncertainties. The uncertainties can be classified in parametric or structured and non-parametric or non-structured. The first ones allow representing the uncertainties into the model coefficients (e.g. the value of a pole placed between maximum and minimum limits). The second ones represent uncertainties as unmodelled dynamics (e.g. differences in the orders of the model and the real system) (Morari and Zafiriou, 1989). The robust control technique which considers more exactly the uncertainties is the Quantitative Feedback Theory (QFT). It is a methodology to design robust controllers based on frequency domain, and was developed by Prof. Isaac Horowitz (Horowitz, 1982; Horowitz and Sidi, 1972; Horowitz, 1993). This technique allows designing robust controllers which fulfil some minimum quantitative specifications considering the presence of uncertainty in the plant model and the existence of perturbations. With this theory, Horowitz showed that the final aim of any control design must be to obtain an open-loop transfer function with the suitable bandwidth (cost of feedback) in order to sensitize the plant and reduce the perturbations. The Nichols plane is used to achieve a desired robust design over the specified region of plant uncertainty where the aim is to design a compensator C(s) and a prefilter F(s) (if it is necessary) (see Figure 1), so that performance and stability specifications are achieved for the family of plants. 392 Robust Control, Theory and Applications This chapter presents for SISO (Single Input Single Output) LTI (Linear Time Invariant) systems, a detailed description of this robust control technique and two real experiences where QFT has successfully applied at the University of Almeria (Spain). It starts with a QFT description from a theoretical point of view, afterwards section 3. 1 is devoted to present two well-known software tools for QFT design, and after that two real applications in agricultural spraying tasks and solar energy are presented. Finally, the chapter ends with some conclusions. 2. Synthesis of SISO LTI uncertain feedback control systems using QFT QFT is a methodology to design robust controllers based on frequency domain (Horowitz, 1993; Yaniv, 1999). This technique allows designing robust controllers which fulfil some quantitative specifications. The Nichols plane is the key tool for this technique and is used to achieve a robust design over the specified region of plant uncertainty. The aim is to design a compensator C(s) and a prefilter F(s) (if it is necessary), as shown in Figure 1, so that performance and stability specifications are achieved for the family of plants p(s) describing a plant P(s). Here, the notation a is used to represent the Laplace transform for a time domain signal a(t). ro *>- i> <> Fig. 1. Two degrees of freedom feedback system. The QFT technique uses the information of the plant uncertainty in a quantitative way, imposing robust tracking, robust stability, and robust attenuation specifications (among others). The 2DoF compensator {F, C}, from now onwards the s argument will be omitted when necessary for clarity, must be designed in such a way that the plant behaviour variations due to the uncertainties are inside of some specific tolerance margins in closed-loop. Here, the family p(s) is represented by the following equation p(s) = [P(s) , rg = i(s + Zi)rELi(s 2 + 2g z c«;o z + a;g z ) > n?=i(s + pt) nti(s 2 + 2ft"o* + ", 2 OtJ max\. k £ [kmin'kmaxl/ z i £ [ z i,miw z i,max\> V? ^ \Vr,miw Vr,n bz £ [<vz,min' bz,max\/ ^Oz £ [^Oz^im ^Oz,max\/ £t € [£,t,min> £t,max]> ^Ot € [ w 0f,min/ ^Ot,max]/ n+m < a+b+ n\ A typical QFT design involves the following steps: a) A Frequency Domain Quantitative Technique for Robust Control System Design 393 1. Problem specification. The plant model with uncertainty is identified, and a set of working frequencies is selected based on the system bandwidth, Q ={coi,co2,---,CL>k)- The specifications (stability, tracking, input disturbances, output disturbances, noise, and control effort) for each frequency are defined, and the nominal plant Pq is selected. 2. Templates. The quantitative information of the uncertainties is represented by a set of points on the Nichols plane. This set of points is called template and it defines a graphical representation of the uncertainty at each design frequency to. An example is shown in Figure 2, where templates of a second-order system given by P(s) = k/s(s + a), with k G [1,10] and a £ [1,10] are displayed for the following set of frequencies Q = {0.5, 1, 2, 4, 8, 15, 30, 60, 90, 120, 180} rad/s. 3. Bounds. The specifications settled at the first step are translated, for each frequency cv in Q set, into prohibited zones on the Nichols plane for the loop transfer function Lq(jco) = C(jcv)Pq(jcv). These zones are defined by limits that are known as bounds. There exist so many bounds for each frequency as specifications are considered. So, all these bounds for each frequency are grouped showing an unique prohibited boundary. Figure 3 shows an example for stability and tracking specifications. TEMPLATES -150 -100 (dB) Fig. 2. QFT Template example. Loop shaping. This phase consists in designing the C controller in such a way that the nominal loop transfer function Lq(jcv) = C(jcv)Pq(jco) fulfils the bounds calculated in the previous phase. Figure 3 shows the design of Lq where the bounds are fulfilled at each design frequency. Prefilter. The prefilter F is designed so that the closed-loop transfer function from reference to output follows the robust tracking specifications, that is, the closed-loop system variations must be inside of a desired tolerance range, as Figure 4 shows. 394 Robust Control, Theory and Applications NICHOLS PLOT ^ :; =;s s : _-'■¥ (m Fig. 3. QFT Bound and Loop Shaping example. PRE-FILTER SHfi Cdl Fig. 4. QFT Prefilter example. A Frequency Domain Quantitative Technique for Robust Control System Design 395 6. Validation. This step is devoted to verify that the closed-loop control system fulfils, for the whole family of plants, and for all frequencies in the bandwith of the system, all the specifications given in the first step. Otherwise, new frequencies are added to the set Q, so that the design is repeated until such specifications are reached. The closed-loop specifications for system in Figure 1 are typically defined in time domain and /or in the frequency domain. The time domain specifications define the desired outputs for determined inputs, and the frequency domain specifications define in terms of frequency the desired characteristics for the system output for those inputs. In the following, these types of specifications are described and the specifications translation problem from time domain to frequency domain is considered. 2.1 Time domain specifications Typically, the closed-loop specifications for system in Figure 1 are defined in terms of the system inputs and outputs. Both of them must be delimited, so that the system operates in a predetermined region. For example: 1. In a regulation problem, the aim is to achieve a plant output close to zero (or nearby a determined operation point). For this case, the time domain specifications could define allowed operation regions as shown in Figures 5a and 5b, supposing that the aim is to achieve a plant output close to zero. 2. In a reference tracking problem, the plant output must follow the reference input with determined time domain characteristics. In Figure 5c a typical specified region is shown, in which the system output must stay. The unit step response is a very common characterization, due to it combines a fast signal (an infinite change in velocity at t = + ) with a slow signal (it remains in a constant value after transitory). The classical specifications such as rise time, settling time and maximum overshoot, are special cases of examples in Figure 5. All these cases can be also defined in frequency domain. 2.2 Frequency domain specifications The closed-loop specifications for system in Figure 1 are typically defined in terms of inequalities on the closed-loop transfer functions for the system, as shown in Equations (2)-(7). 1. Disturbance rejection at the plant output: l + P(ja>)C(jo>) 2. Disturbance rejection at the plant input: c J t 3. Stability: 4. References Tracking: Bi(a>) < P(i">) 1 + P(jco)C(ju) P(jiv)C(jto) l + P(jcv)C(jco) P(jw)P{jco)C(jw < 5 po (co) Vw >0, VPe p < 8 pi {w) Vw > 0, VP 6 < A Vo; > 0, VP G p l + P(ju>)C(ju>) < B u (to)\/iv > 0, VPe p (2) (3) (4) (5) 396 Robust Control, Theory and Applications Allowed operation region (a) Regulation problem (b) Regulation problem for other initial conditions 0.5 (c) Tracking problem Fig. 5. Specifications examples in time domain. 4.5 5 A Frequency Domain Quantitative Technique for Robust Control System Design 397 5. Noise rejection: 6. Control effort: c ft it ft P{jco)C{jw) l + P(ja>)C(ju>) C(jco) l + P(jw)C(ju>) <5„(co)Vw >0, VPe p < Sce(oj) Vw > 0, VP 6 p (6) (7) For specifications in Eq. (2), (3) and (5), arbitrarily small specifications can be achieved designing C so that \C(jco) | — > oo (due to the appearance of the M-circle in the Nichols plot). So, with an arbitrarily small deviation from the steady state, due to the disturbance, and with a sensibility close to zero, the control system is more independent of the plant uncertainty Obviously, in order to achieve an increase in |C(;a?)| is necessary to increase the crossover frequency 1 for the system. So, to achieve arbitrarily small specifications implies to increase the bandwidth 2 of the system. Note that the control effort specification is defined, in this context, from the sensor noise n to the control signal u. In order to define this specification from the reference, only the closed-loop transfer function from the n signal to u signal must be multiplied by F precompensator. However, in QFT, it is not defined in this form because of F must be used with other purposes. On the other hand, to increase the value of \C(jcv)\ implies a problem in the case of the control effort specification and in the case of the sensor noise rejection, since, as was previously indicated, the bandwidth of the system is increased (so the sensor noise will affect the system performance a lot). A compromise must be achieved among the different specifications. The stability specification is related to the relative stability margins: phase and gain margins. Hence, supposing that A is the stability specification in Eq. (4), the phase margin is equal to 2 • arcsin(0.5A) degrees, and the gain margin is equal to 20logio(l + 1/A) dB. The output disturbance rejection specification limits the distance from the open-loop transfer function L(jto) to the point ( — 1,0) in Nyquist plane, and it sets an upper limit on the amplification of the disturbances at the plan output. So, this type of specification is also adequated for relative stability. 2.3 Translation of quantitative specifications from time to frequency domain As was previously indicated, QFT is a frequency domain design technique, so, when the specifications are given in the time domain (typically in terms of the unit step response), it is necessary to translate them to frequency domain. One way to do it is to assume a model for the transfer function T cr , closed-loop transfer function from reference r to the output c, and to find values for its parameters so that the defined time domain limits over the system output are satisfied. 2.3.1 A first-order model Lets consider the simplest case, a first-order model given by T cr (s) - r(t) is an unit step the system output is given by c(t) = (K/a)(l ■ reach c(t) = r(t) for a time t large enough, K should be K = a. K/ (s + a), so that when - e~ at ). Then, in order to 1 The crossover frequency for a system is defined as the frequency in rad/s such that the magnitude of the open-loop transfer function L(jco) — P(jco)C(jco) is equal to zero decibels (dB). 2 The band with of a system is defined as the value of the frequency a^ in rad/s such that \Tcr{j&b) /Tcr{Q)\dB = "3 dB, where T cr is the closed-loop transfer function from the reference r to the output c. 398 Robust Control, Theory and Applications For a first-order model t c = 1/a = l/co^is the time constant (represents the time it takes the system step response to reach 63.2% of its final value). In general, the greater the bandwith is, the faster the system output will be. One important difficulty for a first-order model considered is that the first derivative for the output (in time infinitesimaly after zero, t = + ) is c = K, when it would be desirable to be 0. So, problems appear at the neighborhood of time t = 0. In Figure 6 typical specified time limits (from Eq. (5) Bj and B u are the magnitudes of the frequency response for these time domain limits) and the system output are shown when a first-order model is used. As observed, problems appear at the neighborhood of time t = 0. On the other hand the first-order model does not allow any overshoot, so from the specified time limits the first order model would be very conservative. Hence, a more complex model must be used for the closed-loop transfer function T rr . Fig. 6. Inadequate first-order model. 2.3.2 A second-order model In this case, two free parameters are available (assuming unit static gain): the damping factor £ and the natural frequency oo n (rad/s). The model is given by T(s) s 2 + 2^co n s + col (8) The unit step response, depending on the value of £, is given by A Frequency Domain Quantitative Technique for Robust Control System Design 399 c(t) l-e-^*(cos(w ny /T^t) + - S%_ p sm(u>„y/T=Ft)) if £ < 1 1 _ e -^(cosh(aW£ 2 " If) + g % , smh{w„jl^?t)) if J > 1 w n yg 1 l-e-^(l + oV) if g" = 1 In practice, the step response for a system usually has more terms, but normally it contains a dominant second-order component with £ < 1. The second-order model is very popular in control system design in spite of its simplicity, because of it is applicable to a large number of systems. The most important time domain indexes for a second-order model are: overshoot, settling time, rise time, damping factor and natural frequency In frequency domain, its most important indexes are: resonance peak (related with the damping factor and the overshoot), resonance frequency (related with the natural frequency), and the bandwidth (related with the rise time). The resonance peak is defined as m % x \T cr (jco) | = Mp. The resonance frequency cop is defined as the frequency at which \T cr (jcOp)\ = Mp. One way to control the overshoot is setting an upper limit over Mp. For example, if this limit is fixed on 3 dB, and the practical Tcr(jto) for co in the frequency range of interest is ruled by a pair of complex conjugated poles, then this constrain assures an overshoot lower than 27%. In (Horowitz, 1993) tables with these relations are proposed, where, based on the experience of Professor Horowitz, makes to set a second-order model to be located inside the allowed zone defined by the possible specifications. As Horowitz suggested in his book, if the magnitude of the closed-loop transfer function T cr is located between frequency domain limits B u (co) and B\{co) in Eq. (5), then the time domain response is located between the corresponding time domain specifications, or at most it would be satisfied them in a very approximated way. 2.3.3 A third-order model with a zero A third-order model with a unit static gain is given by T(s) = ^ (9) (s 2 + li^COnS + col) (s + }iCO n ) For values of \i less than 5, a similar behaviour as if the pole is not added to the second-order model is obtained . So, the model in Eq. (8) would must be used. If a zero is added to Eq. (9), it results T(s) = (l + s/^to n )jico 3 n (1Q) (s 2 + 2£,co n s + col) (s + ]ico n ) The unit responses obtained in this case are shown in Figure 7 for different values of A. As shown in Figure 7, this model implies an improvement with respect to that in Eq. (8), because of it is possible to reduce the rise time without increasing the overshoot. Obviously, if co n > 1, then the response is co n times faster than the case with co n = 1 (slower for co n < 1). In (Horowitz, 1993), several tables are proposed relating parameters in Eq. (10) with time domain parameters as overshoot, rise time and settling time. 400 Robust Control, Theory and Applications Fig. 7. Third-order model with a zero for ]i = 5 and f = 1. There exist other techniques to translate specifications from time domain to frequency domain, such as model-based techniques, where based on the structures of the plant and the controller, a set of allowed responses is defined. Another technique is that presented in (Krishnan and Cruickshanks, 1977), where the time domain specifications are formulated as Jo |c(r) — m(r)\ 2 dT < J v 2 (r)dT, with m(t) and v(t) specified time domain functions, and where it is established that the energy of the signal, difference between the system output and the specification m(t), must be enclosed by the energy of the signal v(t), for each instant t, and with a translation to the frequency domain given by the inequality \c(jco) — rh(jco) \ < \v(jcv) | . In (Pritchard and Wigdorowitz, 1996) and (Pritchard and Wigdorowitz, 1997), the relation time-frequency is studied when uncertainty is included in the system, so that it is possible to know the time domain limits for the system response from frequency response of a set of closed-loop transfer functions from reference to the output. This technique may be used to solve the time-frequency translation problem. However, the results obtained in translation from frequency to time and from time to frequency are very conservative. 2.4 Controller design Now, the procedure previously introduced is explained more in detail. The aim is to design the 2DoF controller {F, G} in Figure 1, so that a subset of specifications introduced in section 2.2is satisfied, and the stability of the closed-loop system for all plant P in p is assured. The specifications in section 2. 2are translated in circles on Nyquist plane defining allowed zones for the function L(jcv) = P(jcv)C(jcv). The allowed zone is the outside of the circle for specifications in Eq. (2)-(6), and the inside one for the specification in Eq. (7). Combining the allowed zones for each function L corresponding to each plant P in p, a set of restrictions for controller C for each frequency cv is obtained. The limits of these zones represented in Nichols A Frequency Domain Quantitative Technique for Robust Control System Design 401 plane are called bounds or boundaries. These constrains in frequency domain can be formulated over controller C or over function Lq = PqC, for any plant Pq in p (so-called nominal plant). In order to explain the detailed design process, the following example, from (Horowitz, 1993), is used. Lets suppose the plant in Figure 1 given by P- h(s) = — with/cG [1,20] and a e [1,5]} (11) corresponding to a range of motors and loads, where the equation modeling the motor dynamic is Jc + Be = Ku, with k = K/J and a = B/J in Eq. (11). Lets suppose the tracking specifications given by Bi(cv) < \T cr (jcv)\ dB F(jcv)P(jcv)C(jcv) l + P(jcv)C(jcv) < B u [w) VP e pMco > (12) dB shown in Figure 8. In Figure 9, the difference S(cv) = B u (co) — B/(o?) is shown for each frequency a;. It is easy to see that in order to satisfy the specifications in Eq. (12), the following inequality must be satisfied A\T cr (jcv)\ dB =max Pep P(jcv)C(jcv) l + P(jcv)C(jcv) — mm dB P ^P P(jcv)C(jcv) l + P(jcv)C(jcv) < S(w) = B u (w)-Bi((v)W e pVo; > (13) B (go) enlarged Frequency (rad/s) Fig. 8. Tracking specifications (variations over a nominal). 402 Robust Control, Theory and Applications Frequency (rad/s) Fig. 9. Specifications on the magnitude variations for the tracking problem. Making L = PC large enough, for each plant P in p, and for a frequency to, it is possible to achieve an arbitrarily small specification S(cv). However, this is not possible in practice, since the system bandwidth must be limited in order to minimize the influence of the sensor noise at the plant input. When C has been designed to satisfy the specifications in Eq. (13), the second degree of freedom, F, is used to locate those variations inside magnitude limits B\{co) and B u (cv). In order to design the first degree of freedom, C, it is necessary to define a set of constrains on C or on Lq in the frequency domain, what guarantee that if C (respectively Lq) satisfies those restrictions then the specifications are satisfied too. As commented above, these constrains are called bounds or boundaries in QFT, and in order to compute them it is necessary to take into account: (i) A set of specifications in frequency domain, that in the case of tracking problem, are given by Eq. (13), and that in other cases (disturbance rejection, control effort, sensor noise,...) are similar as shown in section 2.2 (ii) An object (representation) modeling the plant uncertainty in frequency domain, so-called template. The following sections explain more in detail the meaning of the templates and the bounds. Computation of basic graphical elements to deal with uncertainties: templates If there is no uncertainty in plant, the set p would contain only one transfer function, P, and for a frequency, to, P(jcv) would be a point in the Nichols plane. Due to the uncertainty, a set of points, for each frequency, appears in the Nichols plane. One point for each plant P in p. These sets are called templates. For example, Figure 10 shows the template for to = 2 rad/s, corresponding to the set: A Frequency Domain Quantitative Technique for Robust Control System Design 403 D i i i 10 _ k=20 E _ F 5 - . " - CM n CO " -5 - k=2 " -10 - " -15 - / V k=1 " -20 - B - C -155 -150 -145 -140 -135 -130 -125 -120 Angle(P) - degrees Fig. 10. Template for frequency co = 2 rad/s and the plant given by Eq. (11). 3(a; ke [1,20] and a e [1,5] .2/(2;' + a) For k = 1 and driving a from 1 to 5, the segment ABC is obtained in Figure 10. For a = 3 and driving k from 1 to 20, the segment BE is calculated. For k = 20 and driving a from 1 to 5, the segment DEF is obtained. Choosing a plant Pq belonging to the set p, the nominal open-loop transfer function is defined as Lq = PqC. In order to shift a template in the Nichols plane, a quantity must be added in phase (degrees) and other quantity in magnitude (decibels) to all points. Using the nominal point Po(j to) as representative of the full template at frequency co and shaping the value of the nominal Lq(jco) = Pq(jco)C(jco) using C(jco), it is equivalent to add \C(jco)\^ B in magnitude and Angle(C(jco)) degrees in phase to each point P(joo) (with magnitude in decibels and phase in degrees) inside the template at frequency co. So, the shaping of the nominal open-loop transfer function at frequency co (using the degree of freedom C), is equivalent to shift the template at that frequency a; to a specific location in the Nichols plane. The choice of the nominal plant for a template is totally free. The design method is valid independently of this choice. However, there exist rules for the more adequate choice in specific situations (Horowitz, 1993). As was previously indicated, there exists a template for each frequency, so that after the definition of the specifications for the control problem, the following step is to define a set of design frequencies Q. Then, the templates would be computed for each frequency co in Q. Once the specifications have been defined and the templates have been computed, the third step is the computation of boundaries using these graphical objects and the specifications. 404 Robust Control, Theory and Applications Derivation of boundaries from templates and specifications Now, zones on Nichols plane are defined for each frequency cv in Q, so that if the nominal of the template shifted by C(jco) is located inside that zone, then the specifications are satisfied. For each specification in section 2. 2 and for each frequency cv in O, using the template and the corresponding specification, the boundary must be computed. Details about the different types of bounds and the most important algorithms to compute them can be found in (Moreno et al., 2006). In general, a boundary at frequency cv defines a limit of a zone on Nichols plane so that if the nominal Lq(jcv) of the shifted template is located inside that zone, then some specifications are satisfied. So, the most single appearance of a boundary defines a threshold value in magnitude for each phase cp in the Nichols plane, so that if Angle(Lo(jcv)) = cp, then |Lo(;o;)|^g must be located above (or below depending on the type of specification used to compute the boundary) that threshold value. It is important to note that sometimes redefinition of the specifications is necessary. For example, for system in Eq. (11), for cv > 10 rad/s the templates have similar dimensions, and the specifications from Eq. (13) in Figure 9 are identical. Then, the boundaries for cv > 10 rad/s will be almost identical. The function Lq(jcv) must be above the boundaries for all frequencies, including cv > 10 rad/s, but this is unviable due to it must be satisfied that Lq(jcv) — > when cv —> oo. Therefore, it is necessary to open the tracking specifications for high frequency (where furthermore the uncertainty is greater), such as it is shown in Figure 8. On the other hand, it must be also taken into account that for a large enough frequency cv, the specification S(cv) in Eq. (13) must be greater or equal than p|p \P{jcv)\^— p^ \P{jcv)\^ such that, for a small value of Lq(jcv) for these frequencies, the specifications are also satisfied. The effect of this enlargement for the specifcations is negligible when the modifications are introduced at a frequency large enough. These effects are notable in the response at the neighborhood of t = 0. Considering the tracking bounds as negligible from a specific frequency (in the sense that the specification is large enough), it implies that the stability boundaries are the dominant ones at these frequencies. As was mentioned above, since the templates are almost identical at high frequencies and the stability specification A is independent of the frequency, the stability bounds are also identical and only one of them can be used as representative of the rest. In QFT, this boundary is usually called high frequency bound, and it is denoted by B^. Notice that the use of a discrete set of design frequencies Q does not imply any problem. The variation of the specifications and the variation of the appearance of the templates from a frequency cv~ to a frequency cv + , with cv~ < cv < cv+ , is smooth. Anyway, the methodology let us discern the specific cases in which the number of elements of O is insufficient, and let us iterate in the design process to incorporate the boundaries for those new frequencies, then reshaping again the compensator {¥, C}. Design of the nominal open-loop transfer function fulfilling the boundaries In this stage, the function Lq (jcv) must be shaped fulfilling all the boundaries for each frequency. Furthermore, It must assure that the transfer function 1 + L(s) has no zeros in the right half plane for any plant P in p. So, initially Lq = Pq (C = 1) and poles and zeros are added to this function (poles and zeros of the controller C) in order to satisfy all of these restrictions on the Nichols plane. In this stage, only using the function Lq, it is possible to assure the fulfillment of the specifications for all of the elements in the set p when Lq(jcv) is located inside the allowed zones defined by the boundary at frequency cv (computed from the corresponding template at that frequency, and from the specifications). A Frequency Domain Quantitative Technique for Robust Control System Design 405 Obviously, there exists an infinite number of acceptable functions Lq satisfying the boundaries and the stability condition. In order to choose among all of these functions, an important factor to be considered is the sensor noise effect at the plant input. The closed-loop transfer function from noise n to the plant input u is given by Tun(s) -L(s)/P(s) l + P(s)C(s) 1 + L(s) -C(s) In the range of frequencies in which \L(jco)\ is large (generally low frequency), \T un (jco)\ — > \l/P(jco)\, so that the value of \T un (jco)\ at low frequency is independent on the design chosen for L. In the range of frequencies where \L{jco)\ is small (generally high frequency), \T un (jco)\ —> \G(jco)\. These two asymptotes cross between themselves at the crossover frequency. In order to reduce the influence of the sensor noise at the plant input, \C(jco)\ — ► when co — > oo must be guaranteed. It is equivalent to say that \Lq(jco)\ must be reduced as fast as possible at high frequency. A conditionally stable 3 design for Lq is especially adequate to achieve this objective. However, as it is shown in (Moreno et al., 2010) this type of designs supposes a problem when there exists a saturation non-linearity type in the system. Design of the prefilter At this point, only the second degree of freedom, F, must be shaped. The controller C, designed in the previous step, only guarantees that the specifications in Eq. (13) are satisfied, but not the specifications in Eq. (12). Using F, it is possible to guarantee that the specifications in Eq. (12) are satisfied when with C the specifications in Eq. (13) are assured. In order to design F, the most common method consists of computing for each frequency co the following limits Fu(co) max Pep P(jco)C(jco) l + P(jco)C(jco) B u (co) and Fi(co) mm Pep P{jw)C{jw) l + P(ju>)C(jw) B,(to) and shaping F adding poles and zeros until Fi(co) < \F(jco)\ < F u (co) for all frequency co in Q. Validation of the design This is the last step in the design process and consists in studying the magnitude of the different closed-loop transfer functions, checking if the specifications for frequencies outside of the set Q are satisfied. If any specification is not satisfied for a specific frequency, co v , then this frequency is added to the set Q, and the corresponding template and boundary are 3 A system is conditionally stable if a gain reduction of the open-loop transfer function L drives the closed-loop poles to the right half plane. 406 Robust Control, Theory and Applications computed for that frequency to p. Then, the function Lq is reshaped, so that the new restriction is satisfied. Afterwards, the precompensator F is reshaped, and finally the new design is validated. So, an iterative procedure is followed until the validation result is satisfactory. 3. Computer-based tools for QFT As it has been described in the previous section, the QFT framework evolves several stages, where a continuous re-design process must be followed. Furthermore, there are some steps requiring the use of algorithms to calculate the corresponding parameters. Therefore, computer-based tools as support for the QFT methodology are highly valuable to help in the design procedure. This section briefly describes the most well-known tools available in the literature, The Matlab QFT Toolbox (Borghesani et al, 2003) and SISO-QFTIT (Diaz et al., 2005a),(Diaz et al, 2005b). 3.1 Matlab QFT toolbox The QFT Frequency Domain Control Design Toolbox is a commercial collection of Matlab functions for designing robust feedback systems using QFT, supported by the company Terasoft, Inc (Borghesani et al., 2003). The QFT Toolbox includes a convenient GUI that facilitates classical loop shaping of controllers to meet design requirements in the face of plant uncertainty and disturbances. The interactive GUI for shaping controllers provides a point-click interface for loop shaping using classical frequency domain concepts. The toolbox also includes powerful bound computation routines which help in the conversion of closed-loop specifications into boundaries on the open-loop transfer function (Borghesani et al., 2003). The toolbox is used as a combination of Matlab functions and graphical interfaces to perform a complete QFT design. The best way to do that is to create a Matlab script including all the required calls to the corresponding functions. The following lines briefly describe the main steps and functions to use, where an example presented in (Borghesani et al., 2003) is followed for a better understanding (a more detailed description can be found in (Borghesani et al., 2003)). The example to follow is described by: P = { P ^ = ( s + a )( s + b ) : k = [1*2,5,8, 10],« = [1,3,5],*= [20,25,30]}. (14) Once the process and the associated uncertainties are defined, the different steps, explained in section 2 y to design the robust control scheme using the QFT toolbox are described in the following: • Template computation. First, the transfer function models representing the process uncertainty must be written. The following code calculates a matrix of 40 plant elements which is stored in the variable P and represents the system defined by Eq. (14). » c = 1; k = 10; b = 20; » for a = linspace(l,5,10), P(l,l,c) = tf(k,[l,a+b,a*b]); c = c + 1; » end » k = 1; b = 30; » for a = linspace(l,5,10), A Frequency Domain Quantitative Technique for Robust Control System Design 407 P(l,l,c) = tf(k,[l,a+b,a*b]); c = c + 1; » end » b = 30; a = 5; » for k = linspace(l,10,10), P(l,l,c) = tf(k, [l,a+b,a*b]); c = c + 1; » end » b = 20; a = 1; » for k = linspace(l,10,10), P(l,l,c) = tf(k, [l,a+b,a*b]); c = c + 1; » end Then, the nominal element is selected: » nompt=21; and the frequency array is set: » w = [0.1, 5, 10, 100]; Finally, the templates are calculated and visualized using the plottmpl function (see (Borghesani et al., 2003) for a detailed explanation): » plottmpl(w,P,nompt); obtaining the templates shown in Figure 11. 00 Open-Lug Phase- (de^'i Fig. 11. Matlab QFT Toolbox. Templates for example in Eq. (14) • Specifications. In this step, the system specifications must be defined according to Eq. (2)-(7). Once the specifications are determined, the corresponding bounds on the Nichols plane are computed. The following source code shows the use of specifications in Eq. (2)-(4) for this example. A stability specification of A = 1.2 in Eq. (4) corresponding to a gain margin (GM) > 5.3 dB and a phase margin (PM) = 49.25 degrees is given: » Wsl = 1.2; Then, the stability bounds are computed using the function sisobnds (see (Borghesani et al., 2003) for a detailed explanation) and its value is stored in the variable bdbl: 408 Robust Control, Theory and Applications » bdbl = sisobnds(l,w,Wsl,P,0,nompt); Lets now consider the specifications for output and input disturbance rejection cases, from Eq. (2)-(3). For the case of the output disturbance specification, the performance weight for the bandwidth [0,10] is defined as » Ws2 = tf(0.02*[l,64,748,2400],[l,14.4,169]); and the bounds are computed in the following way » bdb2 = sisobnds(2,w(l:3),Ws2,P,0,nompt); For the input disturbance case, the specification is defined as constant for » Ws3 = 0.01; calculating the bounds as » bdb3 = sisobnds(3,w(l:3),Ws3,P,0,nompt); also for the bandwidth [0,10]. Once the specifications are defined and the corresponding bounds are calculated. For each frequency they can be combined using the following functions: » bdb = grpbnds(bdbl,bdb2,bdb3); // Making a global structure » ubdb = sectbnds(bdb); / / Combining bounds The resulting bounds which will be used for the loop-shaping stage are shown in Figure 12. This figure is obtained using the plotbnds function: » plotbnds(ubdb); it v-i :ii-i.yB<M^Ja -225 -IB Fig. 12. Matlab QFT Toolbox. Boundaries for example (14) • Loop-shaping. After obtaining the stability and performance bounds, the next step consists in designing (loop shaping) the controller. The QFT toolbox includes a graphical interactive GUI, Ipshape, which helps to perform this task in an straightforward way. Before using this function, it is necessary to define the frequency array for loop shaping, the nominal plant, and the initial controller transfer function. Therefore, these variables must be set previously, where for this example are given by: » wl = logspace(-2,3,100); // frequency array for loop shaping » CO = tf(l,l); // Initial Controller A Frequency Domain Quantitative Technique for Robust Control System Design 409 » L0=P(l,l,nompt)*C0; // Nominal open-loop transfer function Having defined these variables, the graphical interface is opened using the following line: » lpshape(wl,ubdb,LO,CO); obtaining the window shown in Figure 13. As shown from this figure, the GUI allows to modify the control transfer functions adding, modifying, and removing poles and zeros. This task can be done from the options available at the right area of the windows or dragging interactively on the loop Lq(s) = Pq(s)C(s) represented by the black line on the Nichols plane. For this example, the final controller is given by (Borghesani et al., 2003) C(s) 379(^ + 1) _£_ _|_ _s_ _|_1 247 2 ^ 247 ^ 1 (15) 1-lDlxl Open-Loop: -1.62 deg, -1 7.64 dB Closed-Loop: -1.43 deg, -18.71 <i -lijuin:'/. Varad/sec <? Rad/Sec C Hert: Fig. 13. Matlab QFT Toolbox. Loop shaping for example in Eq. (14) • Pre-filter design. When the control design requires tracking of reference signals, although this is not the case for this example, a pre-filter F(s) must be used in addition to the controller C(s) such as discussed in section 2.. The prefilter can be also designed interactively using a graphical interface similar to that described for the loop shaping stage. To run this option, the pf shape function must be used (see (Borghesani et al., 2003) for more details). • Validation. The control system validation can be done testing the resulting robust controller for all uncertain plants defined by Eq. (14) and checking that the different specifications are fulfilled for all of them. This task can be performed directly programming in Matlab or using the chksiso function from the QFT toolbox. 3.2 An interactive tool based in Sysquake: SISO-QFTIT SISO-QFTIT is a free software interactive tool for robust control design using the QFT methodology (Diaz et al., 2005a;b). The main advantages of SISO-QFTIT compared to other existing tools are its easiness of use and its interactive nature. In the tool described in the previous section, a combination between code and graphical interfaces must be used, where 41 Robust Control, Theory and Applications some interactive features are also provided for the loop shaping and filter design stages. However, with SISO-QFTIT all the stages are available from an interactive point of view. As commented above, the tool has been implemented in Sysquake, a Matlab-like language with fast execution and excellent facilities for interactive graphics (Piguet, 2004). Windows, Mac, and Linux operating systems are supported. Since this tool is completely interactive, one consideration that must be kept in mind is that the tool's main feature -interactivity- cannot be easily illustrated in a written text. Thus, the reader is cordially invited to experience the interactive features of the tool. The users mainly should operate with only mouse operations on different elements in the window of the application or text insertion in dialog boxes. The actions that they carry out are reflected instantly in all the graphics in the screen. In this way the users take aware visually of the effects that produce their actions on the design that they are carrying out. This tool is specially conceived as much as for beginner users that want to learn the QFT methodology, as for expert users (Diaz et al., 2005b). The user can work with SISO-QFTIT in two different but not excluding ways (Diaz et al., 2005b): • Interactive mode. In this work form, the user selects an element in the window and drags it to take it to a certain value, their actions on this element are reflected simultaneously on all the present figures in the window of the tool. • Dialogue mode. In this work form, the user should simply go selecting entrances of the Settings menu and correctly fill the blanks of dialog boxes. Such as commented in the manual of this interactive software tool, its main interactive advantages and options are the following (Diaz et al., 2005b): • Variations that take place in the templates when modifying the uncertainty of the different elements of the plant or in the value of the template calculation frequency. • Individual or combined variation on the bounds as a result of the configuration of specifications, i.e., by adding zeros and poles to the different specifications. • The movement of the controller zeros and poles over the complex plane and the modification of its symbolic transfer function when the open loop transfer function is modified in the Nichols plane. • The change of shape of the open loop transfer function in the Nichols plane and the variation of the expression of the controller transfer function when any movement, addition or suppression of its zeros or poles in the complex plane. • The changes that take place in the time domain representation of the manipulated and controlled variables due to the modification of the nominal values of the different elements of the plant. • The changes that take place in the time domain representation of the manipulated and controlled variables due to the introduction of a step perturbation at the input of the plant. The magnitude and the occurrence instant of the perturbation is configured by the user by means of the mouse. Such as pointed out above, the interactive capabilities of the tool cannot be shown in a written text. However, some screenshots for the example used with the Matlab QFT toolbox are provided. Figure 14a shows the resulting templates for the process defined by Eq. (14). A Frequency Domain Quantitative Technique for Robust Control System Design 411 Notice that with this tool, the frequencies, the process uncertainties and the nominal plant can be interactively modified. The stability bounds are shown in Figure 14b. The radiobuttons available at the top-right side of the tool allow to choose the desired specification. Once the specification is selected, the rest of the screen is changed to include the specification values in an interactive way. Figure 15a displays the loop shaping stage with the combination of the different bounds (same result than in Figure 13). The figure also shows the resulting loop shaping for controller (15). Then, the validation screen is shown in Figure 15b, where it is possible to check interactively if the robust control design satisfies the specifications for all uncertain cases. Although for this example it is not necessary to design the pre-filter for the tracking specifications, this tool also provides a screen where it is possible to perform this task (see an example in Figure 16). i| ||EH|ft-fl|S| IH^|R^--1 -J 1- -4 A- IT © (a) QFT Templates (b) Stability bounds Fig. 14. SISO-QFTIT. Templates and bounds for the example described in Eq. (14) -i ir^i^i ~T 8= sd 'l • •V, 0/m (a) Loop shaping (b) Validation Fig. 15. SISO-QFTIT. Loop shaping and validation for the example described in Eq. (14) 4. Practical applications This section presents two industrial projects where the QFT technique has been successfully used. The first one is focused on the pressure control of a mobile robot which was design 412 Robust Control, Theory and Applications Fig. 16. SISO-QFTIT. Prefilter stage for spraying tasks in greenhouses (Guzman et al., 2008). The second one deals with the temperature control of a solar collector field (Cirre et al., 2010). 4.1 In agricultural and robotics context: Fitorobot During the last six years, the Automatic Control, Robotics and Electronics research group and the Agricultural Engineering Department, both from the University of Almeria (Spain), have been working in a project aimed at designing, implementation, and testing a multi-use autonomous vehicle with safe, efficient, and economic operation which moves through the crop lines of a greenhouse and which performs tasks that are tedious and /or hazardous for people. This robot has been called Fitorobot. The first version of this vehicle has been equipped for spraying activities, but other configurations have also been designed, such as: a lifting platform to reach high zones to perform tasks (staking, cleaning leaves, harvesting, manual pollination, etc.), and a forklift to transport and raise heavy materials (Sanchez-Gimeno et al., 2006). This mobile robot was designed and built following the paradigm of Mechatronics such as described in (Sanchez-Hermosilla. et al., 2010). The first objective of the project consisted of developing a prototype to enable the spraying of a certain volume of chemical products per hectare while controlling the different variables that affect the spraying system (pressure, flow, and travel speed). The pressure is selected and the control signal keeps the spraying conditions constant (mainly droplet size). The reference value of the pressure is calculated based on the mobile robot speed and the volume of pesticide to apply, where the pressure working range is between 5 and 15 bar. There are some circumstances where it is impossible to maintain a constant velocity due to the irregularities of the soil, different slopes of the ground, and the turning movements between the crop lines. Thus, for work at a variable velocity (Guzman et al., 2008), it is necessary to spray using a variable-pressure system based on the vehicle velocity, which is the proposal adopted and implemented in this work. This system presents some advantages, such as the higher quality of the process, because the product sprayed over each plant is optimal. Furthermore, this system saves chemical products because an optimal quantity is sprayed, reducing the environmental impact and pollution as the volume sprayed to the air is minimized. The robot prototype (Figure 17) consists of an autonomous mobile platform with a rubber tracked system and differential guidance mechanism (to achieve a more homogeneous A Frequency Domain Quantitative Technique for Robust Control System Design 413 distribution of soil-compaction pressure, thus disturbing less the sandy soil typical of Mediterranean greenhouses (Sanchez-Gimeno et al., 2006)). The robot is driven by hydraulic motors fed by two variable displacement pumps powered by a 20-HP gasoline motor, allowing a maximum velocity of 2.9 m/s. Due to the restrictions imposed by the narrow greenhouse lanes, the vehicle dimensions are 70 cm width, 170 cm length, and 180 cm height at the top of the nozzles. Fig. 17. Mobile robot for agricultural tasks ON / OFF ELECTROVALVE PIPES WITH PRESSURE GROUP OF PRESSURE ENGINE PIPES WITHOUT PRESSURE Fig. 18. Scheme of the spraying system 414 Robust Control, Theory and Applications The spraying system carried out by the mobile robot is composed with a 300 1 tank used to store the chemical products, a vertical boom sprayer with 10 nozzles, an on /off electrovalve to activate the spraying, a proportional electrovalve to regulate the output pressure, a double-membrane pump with pressure accumulator providing a maximum flow of 30 1/min and a maximum pressure of 30 bar, and a pressure sensor to close the control loop as shown in Figure 18. In this case, the control problem was focused on regulating the output pressure of the spraying system mounted on the mobile robot despite changes in the vehicle velocity and the nonlinearities of the process. For an adequate control system design, it was necessary to model the plant by obtaining its associated parameters. Several open-loop step-based tests were performed varying the valve aperture around a particular operating point. The results showed that the system dynamics can be approximated by a first-order system with delay. Thus, it can be modelled using the following transfer function P(s) (16) TS + 1 where k is the static gain, t r is the delay time, and t is the time constant. Then, several experiments in open loop were performed to design the dynamic model of the spraying system using different amplitude opening steps (5% and 10%) over the same operating points (see Figure 19a). The analysis of the results showed that the output-pressure behavior changes when different valve-amplitude steps are produced around the same working point, and also when the same valve opening steps are produced at several operating points, confirming the uncertainty and nonlinear characteristics of the system. Opening 5 % Filtered signal "i ! \& s 1G z a 9 10 £ \\ L u i " ^ _-^ i 50 100 150 200 250 300 350 400 450 Time (s) (a) Time domain (b) Frequency domain Fig. 19. System uncertainties from the time and frequency domains After analyzing the results (see Figure 19a), the system was modelled as a first-order dynamical system with uncertain parameters, where the reaction curve method has been used at the different operating points. Therefore, the resulting uncertain model is given by the following transfer function (see Figure 19b): {P(s TS + - : k e [-0.572, -0.150], r e [0.4,1]} (17) A Frequency Domain Quantitative Technique for Robust Control System Design 415 where the gain, k, is given in bar/% aperture and the constant time, t , in seconds. Once the system was characterized, the robust control design using QFT was performed considering specifications on stability and tracking. First, the specifications for each frequency were defined, and the nominal plant Pq was selected. The set of frequencies and the nominal plant were set to Q = {0.1, 1,2, 10} rad/s and Pq = oTi^PT' respectively The stability specification was set to A = 1.2 corresponding to a GM > 5.3 dB and a PM = 49.25, and for the tracking specifications the maximum and minimum values for the magnitude have been described by the following transfer functions (frequency response for tracking specifications are shown in Figure 19b in dashed lines) Bi(s) 10 B u (s) 12.25 (18) s + 10' ww s 2 + 8.75s + 12.25 Figure 20a shows the different templates of the plant for the set of frequencies determined above. i '• i < : o.j I r i i i %£ -1- o° : -$"% : c": ;""^y 2 I ° jo | "! | o o . j.__P___; 9 \ o_; i---S-o- } \ -^~ :I In 3- :: _„i l^r.-^y^^r^T^r^r^^}!^?L £ ; ; /"f . 1 ; ; 1t\ ; I > 1 ) Itl f"\ Phase Idegreesl (a) QFT Templates (b) Loopshaping stage Fig. 20. Templates and feedback controller design by QFT The specifications are translated to the boundaries on the Nichols plane for the loop-transfer function L(jco) = C(jw)P(j to). Figure 20b shows the different bounds for stability and tracking specifications set previously. Then, the loop shaping stage was performed in such a way that the nominal loop-transfer function Lq(jcv) = C(jcv)Pq(jcv) was adjusted to make the templates fulfil the bounds calculated in the previous phase. Figure 20fr shows the design of Lq where the bounds axe fulfilled at each design frequency. This figure shows the optimal controller using QFT to lie on the boundaries at each frequency design. However, a simpler controller fulfilling the specifications was preferred for practical reasons. The resulting controller was the following: C(s) 27.25(s + 1) (19) To conclude the design process, the prefilter F is determined so that the closed-loop transfer function matches the robust tracking specifications, that is, the closed-loop system variations must be inside of a desired tolerance range: F(s) 0.1786s + 1 (20) 416 Robust Control, Theory and Applications Once the robust design was performed, the system was validated by simulation. Figure 21 shows the validation results where the specifications are clearly satisfied for the whole family of plants described by Eq. (17) for the time domain and frequency domain, respectively (a) Time domain (b) Frequency domain Fig. 21. Validation for the QFT design of the pressure system If the results shown in Figure 19b are compared with those shown in Figure 21b, a considerable uncertainty reduction can be appreciated, especially in the gain system. Notice that Figure 19 shows the responses of the open-loop system against step inputs for the time and frequency domains, respectively. From these figures, the system uncertainties can be observed by deviations in the static gain and in the time constant of the system, such as described in equation (17). Finally, the proposed control scheme was tested on the spraying system. The robust control system is characterized by the ability of the closed-loop system to reach desired specifications satisfactorily despite of large variations in the (open-loop) plant dynamics. As commented above, in the pressure system presented in this work such variations appear along the different operating points of the process. Therefore, the system was initially tested through a group of different steps in order to verify that the control system fulfills the robust specifications. Figure 22 shows the results for a sequence of typical steps. It can be observed that the system faithfully follows the proposed reference, reaching the same performance for the different operating points. 4.2 In solar energy field: ACUREX This section presents a robust control scheme for a distributed solar collector (DSC) field. As DSC are systems subjected to strong disturbances (mainly in solar radiation and inlet oil temperature), a series feedforward was used as a part of the plant, so that the system to be controlled has one input (fluid flow) and one output (outlet temperature) as the disturbances are partially compensated by the series feedforward term, so that the nonlinear plant is transformed into an uncertain linear system. The QFT technique (QFT) was used to design a control structure that guarantee desired control specifications, as settling time and maximum overshoot, under different operating conditions despite system uncertainties and disturbances (Cirre et al., 2010). The main difference between a conventional power plant and a solar plant is that the primary energy source, while being variable, cannot be manipulated. The objective of the A Frequency Domain Quantitative Technique for Robust Control System Design 417 14 cc 12 3 10 (/> £ 8 - - - Set-point ^— ^— Pressure 50 100 150 Time (s) 200 250 150 Time (s) Fig. 22. Experimental tests for the spraying system control system in a distributed solar collector field (DCS) is to maintain the outlet oil temperature of the loop at a desired level in spite of disturbances such as changes in the solar irradiance level (caused by clouds), mirror reflectivity, or inlet oil temperature. The means available for achieving this is via the adjustment of the fluid flow and the daily solar power cycle characteristics are such that the oil flow has to change substantially during operation. This leads to significant variations in the dynamic characteristics of the field, which cause difficulties in obtaining adequate performance over the operating range with a fixed parameter controller (Camacho et al., 1997; 2007a;b). For that reason, this section summarizes a work developed by the authors where a robust PID controller is designed to control the outlet oil temperature of a DSC loop using the QFT technique. In this work, the ACUREX thermosolar plant was used, which is located at the Plataforma Solar de Almeria (PSA), a research centre of the Spanish Centro de Investigaciones Energeticas Medioambientales y Tecnologicas (CIEMAT), in Almeria, Spain. The plant is schematically composed of a distributed collector field, a recirculation pump, a storage tank and a three-way valve, as shown in Figures 23 and 24. The distributed collector field consists of 480 east-west-aligned single-axis-tracking parabolic trough collectors, with a total mirror aperture area of 2672 m2, arranged in 20 rows forming 10 parallel loops (see Figure 23). The parabolic mirrors in each collector concentrate the solar irradiation on an absorber tube through which Santotherm 55 heat transfer oil is flowing. For the collector to concentrate sunlight on its focus, the direct solar radiation must be perpendicular to the mirror plane. Therefore, a sun-tracking 418 Robust Control, Theory and Applications algorithm causes the mirrors to revolve around an axis parallel to the tube. Oil is recirculated through the field by a pump that under nominal conditions supplies the field at a flow rate of between 2 1/s (in some applications 3 1/s) and 12 1/s. As it passes through the field, the oil is heated and then the hot oil enters a thermocline storage tank, as shown in Figure 24. A complete detailed description of the ACUREX plant can be found in (Camacho et al., 1997). Fig. 23. ACUREX solar plant OH. THERMAL STORAGE TANK &^ SOLAR COLLECTOR MKI.I) Fig. 24. Simplified layout of the ACUREX plant As described in (Camacho et al., 1997), DSC dynamics can be approximated by low-order linear descriptions of the plant (as is usually done in the process industry) to model the system around different operating conditions and to design diverse control strategies without accounting for system resonances (Alvarez et al., 2007; Camacho et al., 1997). Thus, different low-order models are found for different operating points mainly due to fluid velocity and system disturbances. Using the series feedforward controller (presented in (Camacho et al., 1997) and improved in (Roca et al., 2008)), a nonlinear plant subjected to disturbances is treated as an uncertain linear plant with only one input (the reference temperature to the feedforward controller, T r ff)- After performing an analysis of the frequency response (Berenguel et al., 1994), it was observed that the characteristics of the system (time constants, gains, resonance modes, ...) depend on the fluid flow rate as expected (Alvarez et al., 2007; Camacho et al., 1997). Therefore, in order to control the system with a fixed-parameter controller, the following model has been used P = \P(s) = n }^ n ne~ XdS ■ I = 0.8, (21) T d = 39s, cv n e [0.0038, 0.014]rad/s, k G [0.7,1.05]}, where the chosen nominal plant is Pq(s) with co n = 0.014 rad/s and k = 0.7. A Frequency Domain Quantitative Technique for Robust Control System Design 419 Thus, once the uncertain model has been obtained, the specifications were determined on time domain and translated into the frequency domain for the QFT design. In this case, the tracking and stability specifications were established (Horowitz, 1993). For tracking specifications only is necessary to impose the minimum and maximum values for the magnitude of the closed-loop transfer function from the reference input to the output in all frequencies. With respect to the stability specification, the desired gain (GM) and phase (PM) margins are set. The tracking specifications were required to fulfill a settling time between 5 and 35 minutes and an overshoot less than 30% after 10-20°C setpoint changes for all operating conditions (realistic specifications, see (Camacho et al., 2007a;b)). For stability specification, A = 3.77 in Eq. (4) is selected in order to guarantee at least a phase margin of 35 degrees for all operating conditions. To design the compensator C(s), the tracking specifications in Eq. (13), shown in Table 1 for each frequency in the set of design frequencies Q, are used Table 1. Tracking specifications for the C compensator design co (rad/s) 0.0006 0.001 0.003 0.01 5{co) 0.55 1.50 9.01 19.25 The resulting compensator C(s), synthesized in order to achieve the stability specifications and the tracking specifications previously indicated, is the following PID-type controller (22) C(s)=0 - 75 v 1 + i8^ +40S which represents the resulting loop shaping in Figure 25. Then, in order to satisfy the tracking specifications, the prefilter F(s) must be designed, where the synthesized prefilter is given by F ^ = OT (23) ir | -10 -?z -?.z ^_ — — — ^^_^ y^ / ^Vf j .-. :. IV ■250 -200 -1SO Phase (degrees) Fig. 25. Tracking and stability boundaries with the designed Lq(joj) Figure 26 shows that the tracking specifications are fulfilled for all uncertain cases. Note that the different appearance of Bode diagrams in closed loop for five operating conditions is due to the changing root locus of L(s) when the PID is introduced. 420 Robust Control, Theory and Applications Fraquaney (radys*c> Fig. 26. Tracking specifications (dashed-dotted) and magnitude Bode diagram of some closed loop transfer functions In order to prove the fulfillment of the tracking and stability specifications of the control structure, experiments were performed under several operating points and under different conditions of disturbances (Cirre et al., 2010), although only representative results are shown in this work. Figure 27 shows an experiment with the robust controller. At the beginning of the experiment, the flow is saturated until the outlet temperature is higher than the inlet one (the normal r 3QQ -f— ] r— ! r- 9,5 10,0 10,5 11,0 11,5 12,0 12,5 13,0 13,5 14 : 14 : 5 15 f 15,5 16,0 Local timefh) Fig. 27. QTF control results for the ACUREX plant (24/03/2009) (Cirre et al, 2010) situation during the operation). This situation always appears due to the oil resident inside the pipes is cooler than the oil from the tank. Once the oil is mixed in the pipes, the outlet A Frequency Domain Quantitative Technique for Robust Control System Design 421 temperature reaches a higher temperature than the inlet one. During the start up, steps in the reference temperature are made until reaching the nominal operating point. The overshoot at the end of this phase is 18 °C approximately, and thus the specifications are fulfilled. Analyzing the time responses, a settling time between 11 and 15 minutes is observed at the different operating points. Therefore, both time specifications, overshoot and settling time are properly fulfilled. Disturbances in the inlet temperature (from the beginning until t = 12.0 h), due to the temperature variation of the stratified oil inside the tank, are observed during this experiment and correctly rejected by the feedforward action (Cirre et al., 2010). 5. Conclusions This chapter has introduced the Quantitative Feedback Theory as a robust control technique based on the frequency domain. QFT is a powerful tool which allows to design robust controllers considering the plant uncertainty, disturbances, noise and the desired specifications. It is very versatile tool and has been used in multiple control problems including linear (Horowitz, 1963), non-linear (Moreno et al., 2010), (Moreno et al., 2003), (Moreno, 2003), MIMO (Horowitz, 1979) and non-minimum phase (Horowitz and Sidi, 1978). After describing the theoretical aspects, the most well-known software tools to work with QFT have been described using simple examples. Then, results from two experimental applications were presented, where QFT were successfully used to compensate for the uncertainties in the processes. 6. References J.D. Alvarez, L. Yebra, and M. Berenguel. Repetitive control of tubular heat exchangers. Journal of Process Control, 17:689-701, 2007. M. Berenguel, E.F. Camacho, and F.R. Rubio. Simulation software package for the acurex field. Technical report, Dep. Ingenieria de Sistemas y Automatica, University of Seville (Spain), 1994. www.esi2.us.es/ rubio/ libro2.html. C. Borghesani, Y. Chait, and O. Yaniv. The QFT Frequency Domain Control Design Toolbox. Terasoft, Inc., http://www.terasoft.com/qft/QFTManual.pdf, 2003. E.F. Camacho, M. Berenguel, and F.R. Rubio. Advanced Control of Solar Plants (1st edn). Springer, London, 1997. E.F. Camacho, F.R. Rubio, M. Berenguel, and L. Valenzuela. A survey on control schemes for distributed solar collector fields, part i: modeling and basic control approaches. Solar Energy, 81:1240-1251, 2007a. E.F. Camacho, F.R. Rubio, M. Berenguel, and L. Valenzuela. A survey on control schemes for distributed solar collector fields, part ii: advances control approaches. Solar Energy, 81:1252-1272, 2007b. M.C. Cirre, J.C. Moreno, M. Berenguel, and J.L. Guzman. Robust control of solar plants with distributed collectors. In IFAC International Symposium on Dynamics and Control of Process Systems, DYCOPS, Leuven, Belgium, 2010. J. M. Diaz, S. Dormido, and J. Aranda. Interactive computer-aided control design using quantitative feedback theory: The problem of vertical movement stabilization on a high-speed ferry. International Journal of Control, 78:813-825, 2005a. J. M. Diaz, S. Dormido, and J. Aranda. SISO-QFTIT An interactive software tool for the design of robust controllers using the QFT methodology. UNED, http://ctb.dia.uned.es/asig/qftit/, 2005b. 422 Robust Control, Theory and Applications J.L. Guzman, Rodriguez R, Sanchez-Hermosilla J., and M. Berenguel. Robust pressure control in a mobile robot for spraying tasks. Transactions of the AS ABE, 5 1(2): 71 5-727, 2008. I. Horowitz. Synthesis of Feedback Systems. Academic Press, New York, 1963. I. Horowitz. Quantitative feedback theory. IEEE Proc, 129 (D-6):215-226, 1982. I. Horowitz and M. Sidi. Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances. International Journal of Control, 16 (2):287-309, 1972. I. M. Horowitz. Quantitative Feedback Design Theory (QFT). QFT Publications, Boulder, Colorado, 1993. I.M. Horowitz. Quantitative synthesis of uncertain multiple input-output feedback systems. International Journal of Control, 30:81-106, 1979. I.M. Horowitz and M. Sidi. Optimum synthesis of non-minimum phase systems with plant uncertainty. International Journal of Control, 27(3):361-386, 1978. K. R. Krishnan and A. Cruickshanks. Frequency domain design of feedback systems for specified insensitivity of time-domain response to parameter variations. International Journal of Control, 25 (4):609-620, 1977. M. Morari and E. Zafiriou. Robust Process Control. Prentice Hall, 1989. J. C. Moreno. Robust control techniques for systems with input constrains, (in Spanish, Control Robusto de Sistemas con Restricciones a la Entrada). PhD thesis, University of Murcia, Spain (Universidad de Murcia, Espaha), 2003. J. C. Moreno, A. Banos, and M. Berenguel. A synthesis theory for uncertain linear systems with saturation. In Proceedings of the 4th IFAC Symposium on Robust Control Design, Milan, Italy, 2003. J. C. Moreno, A. Bahos, and M. Berenguel. Improvements on the computation of boundaries in qft. International Journal of Robust and Nonlinear Control, 16(12) -.575-597, May 2006. J. C. Moreno, A. Bahos, and M. Berenguel. A qft framework for anti-windup control systems design. Journal of Dynamic Systems, Measurement and Control, 132(021012):15 pages, 2010. Y. Piguet. Sysquake 3 User Manual. Calerga Sari, Lausanne, Switzerland, 2004. C. J. Pritchard and B. Wigdorowitz. Mapping frequency response bounds to the time domain. International Journal of Control, 64 (2):335-343, 1996. C. J. Pritchard and B. Wigdorowitz. Improved method of determining time-domain transient performance bounds from frequency response uncertainty regions. International Journal of Control, 66 (2):311-327, 1997. L. Roca, M. Berenguel, L.J. Yebra, and D. Alarcon. Solar field control for desalination plants. Solar Energy, 82:772-786, 2008. A. Sanchez-Gimeno, Sanchez-Hermosilla J., Rodriguez E, M. Berenguel, and J.L. Guzman. Self-propelled vehicle for agricultural tasks in greenhouses. In World Congress - Agricultural Engineering for a better world, Bonn, Germany, 2006. J. Sanchez-Hermosilla, Rodriguez E, Gonzalez R., J.L. Guzman, and M. Berenguel. A mechatronic description of an autonomous mobile robot for agricultural tasks in greenhouses. In Alejandra Barrera, editor, Mobile Robots Navigation, pages 583-608. In-Tech, 2010. ISBN 978-953-307-076-6. O. Yaniv, Quantitative Feedback Design of Linear and Nonlinear Control Systems. Kluwer Academic Publishers, 1999. 18 Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics Dr. Sabato Manfredi Faculty of Engineering, University of Naples Federico II, Via Claudio 21, Napoli 80120. Italy 1. Introduction Many systems in nature and of practical interest can be modeled as large collections of interacting subsystems. Such systems are referred as "Multi Agent Systems" (briefly MASs) and some examples include electrical power distribution networks (P. Kundur, 1994), communication (F. Paganini, 2001), and collections of vehicles traveling in formation (J.K. Hedrick et al., 1990). Several practical issues concern the design of decentralized controllers and the stability analysis of MASs in the presence of uncertainties in the subsystem interconnection topology (i.e. due in practical applications to failures of transmission lines). The analysis and control of collections of interconnected systems have been widely studied in the literature. Early work on stability analysis and decentralized control of large-scale interconnected systems is found in (D. Limebeer & Y.S. Hung, 1983; A. Michel & R. Miller, 1977; P.J. Moylan & D.J. Hill, 1978; Siljak, 1978; J.C. Willems, 1976). Some of the more widely notable stability criteria are based on the passivity conditions (M. Vidyasagar, 1977) and on the well-known notion of connective stability introduced in (Siljak, 1978). More recently, MASs have appeared broadly in several applications including formation flight, sensor networks, swarms, collective behavior of flocks (Savkin, 2004; C.C. Cheaha et al., 2009; W. Ren, 2009) motivating the recent significative attention of the scientific community to distributed control and consensus problems (i.e. (R.O. Saber & R. Murray, 2004; Z. Lin et al., 2004; V. Blondel et al., 2005; J. N. Tsitsiklis et al., 1986)). One common feature of the consensus algorithm is to allow every agent automatically converge to a common consensus state using only local information received from its neighboring agents. "Consensusability" of MASs is a fundamental problem concerning with the existence conditions of the consensus state and it is of great importance in both theoretical and practical features of cooperative protocol (i.e. flocking, rendezvous problem, robot coordination). Results about consensuability of MASs are related to first and second order systems and are based on the assumption of jointly-connected interaction graphs (i.e. in (R.O. Saber & R. Murray, 2004; J. N. Tsitsiklis et al., 1986)). Extension to more general linear MASs whose agents are described by LTI (Linear Time Invariant) systems can be found in (Tuna, 2008) where the closed-loop MASs were shown to be asymptotic consensus stable if the topology had a spanning tree. In (L. Scardovi & R. Sepulchre, 2009) it is investigated the synchronization of a 424 Robust Control, Theory and Applications network of identical linear state-space models under a possibly time-varying and directed interconnection structure. Many investigations are carried out when the dynamic structure is fixed and the communication topology is time varying (i.e. in (R.O. Saber & R. Murray, 2004; W. Ren & R. W. Beard, 2005; Ya Zhanga & Yu-Ping Tian, 2009)). One of main appealing field of research is the investigation of the MASs consensusability under both the dynamic agent structure and communication topology variations. In particular, it is worth analyzing the joint impact of the agent dynamic and the communication topology on the MASs consensusability. The aim of the chapter is to give consensusability conditions of LTI MASs as function of the agent dynamic structure, communication topology and coupling strength parameters. The theoretical results are derived by transferring the consensusability problem into the robust stability analysis of LTI-MASs. Differently from the existing works, here the consensuability conditions are given in terms of the adjacency matrix rather than Laplacian matrix. Moreover, it is shown that the interplay among consensusability, node dynamic and topology must be taken into account for MASs stabilization: specifically, consensuability of MASs is assessed for all topologies, dynamic and coupling strength satisfying a pre-specified bound. From the practical point of view the consensuability conditions can be used for both the analysis and planning of MASs protocols to guarantee robust stability for a wide range of possible interconnection topologies, coupling strength and node dynamics. Also, the number of subsystems affecting the overall system stability is taken into account as it is analyzed the robustness of multi agent systems if the number of subsystems changes. Finally, simulation examples are given to illustrate the theoretical analysis. 2. Problem statement We consider a network composed of linear systems interconnected by a specific topological structure. The dynamical system at each node is of m-th order and described by the matrices (A, B, C).Let G(V, E, U) be a directed weighted graph (digraph) with the set of nodes V = l..ft, set of edges E C n x n, and the associated weighted adjacency matrix U = {ujj} with Ujj > if there is a directed edge of weight Uu from vertex j (node parent) into vertex i (node child). The linear systems are interconnected by a directed weighted graph G(V, E, 17). Each node dynamical is described by: ±i(t) = Axi(t) + Bvi(t) Vi {t) = Cxiit) (1) with V((t) is the input to the i-th node of the form *i(0 = £>#/('). (2) In this way, each node dynamic is influenced by the sum of its neighbors' outputs. This yields to the MAS network equation: n ±i(t) = Ax{(t) + £ uijBCxj(t) (3) 7=1 with 1 < i < n, and its compact form: x(t) = Agx{t) (4) Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 425 with Ag = (I n A) + (U BC), with denotes the matrix Kronecker product. Notice that the above equation can be associated to the main model used in the literature for describing the synchronization phenomena, energy distribution, tanks network (e.g. in (R. Cogill & S. Lall, 2004)). Moreover the system at each node can be MIMO or SISO type, and the matrix product BC takes into account the coupling strength and the coupling interaction among the state system variables. Observing the MAS model (3) we point out as the overall network dynamic is affected by the node system dynamic matrix A, the coupling matrix BC, and by the adjacency matrix U of the topological structure. Consider a network with n agents whose topology information exchange is described by a graph G(V, E, U) and let X( the state of agent-node i-th, consensus corresponds to the network condition such that the state of the agents as a whole asymptotically converges to an equilibrium state with identical elements (i.e. Xj = Xj for all (i,j) € n x n). The common value x is named consensus value. Consensusability of MASs is a fundamental problem concerning with the conditions for assessing network consensus equilibrium. Under the assumption of the existence of a network equilibrium, then consensuability deals with the research of analytical conditions such that the network equilibrium corresponds to a consensus state. In this way, without loss of generality, the consensuability problem can be reduced to the problem of assessing stabilization conditions of the MAS network (3) with respect to the equilibrium point (i.e. x\ = Xj = for all (i,j) G n x n). Hence, we are interested in solving the following problem: Problem Given a multi agent network described by (3), to determinate the MAS consensuability conditions as function of node dynamic, topology and coupling strength. Specifically, consensuability of MASs is assessed for all topologies, dynamic and coupling strength satisfying a pre-specified bound. In the follows we will present analytical conditions for solving the above Problem. 3. Conditions for MASs consensuability Before of presenting the MASs consensuability conditions of (3), we have to recast the eigenvalues set cr(Ag) of MAS network dynamic matrix Ag. Lemma 1 Let c((i)={^ z } the eigenvalues set of the adjacency matrix U, cr(A g ) the eigenvalues set of the MAS dynamical matrix Ag, then results: (r(Ag) = |J Z - o~( A + f/ z BC) for all 1 < i < n. Proof Let / the Jordan canonical form of U, then it exists a similarity matrix S so that / = S~ 1 US. Hence S I n is a similarity matrix for the matrices I m A + U BC and I m A + / BC. From the Kronecker product (Horn R.A. & Johnson C.R., 1995) results: (S l n )~ l {lm A + U BC)(S I n ) = (S" 1 I n ) (I m A + U BC) (S I n ) = (I m A) + (S^US BC) = (I m A) + (/ BC) with / being an upper triangular matrix with I m A + / BC as upper triangular block matrix. Hence the eigenvalues of the matrix I m A + / BC are the union of the eigenvalues of the block matrix on the diagonal. 426 Robust Control, Theory and Applications From the above Lemma 1, the eigenvalues of the MAS dynamic matrix Ag are explicitly function of those of the matrix A + jijBC, for all i. So we can decouple the effects of topology structure (by ]i{), the coupling strength BC and node dynamic A on the overall stability of the MAS. This can be used for giving stability MAS condition as function of topology structure, node dynamic and coupling strength as shown by the following Theorem 1: Theorem 1 Let the MAS composed of n identical MIMO system of order m-th and interconnected by the digraph G = (y,E,U) with adjacency matrix U, with eigenvalues Wi < V-2 < • • • Pn- If the node dynamic matrix A = {a^} and the coupling matrix BC = {c z y} fulfill the conditions: *ii + Hk c ii < ( 5 ) V* = 1, 2, ..., m and VA: = 1, 2, ...., ft, then the MAS (3) is stable. Proof If the conditions (5) hold, then all eigenvalues of the matrix A + ]i k BC I an + fi k c n a 12 + \iyc\i • • • a \m + fVin \ «21 + w fc c 21 #22 + ^fcC 2 2 ... «2m + fV 2m \ # m i + fl] c C m i a m 2 + }i-k c m2 • • • %m + }^k c mm / \/k = 1, 2, . . . , n, are located in a convex set in the left complex half plane as result by the application of the Gershgorin's circle theorem (Horn R.A. & Johnson C.R., 1995). Hence, by Lemma 1, the MAS is stable. The previous Theorem 1 easily yields to the following corollaries. Corollary 1 Let the MAS composed of n identical MIMO system of order 2 and interconnected by the digraph G = (V,E,U), with adjacency matrix U with eigenvalues ^1 < ^2 < • • • V-n- If the node dynamic matrix A = {««} and the coupling matrix BC = {c«} with c« > 0, i,j = 1,2, fulfill the conditions: "ij ^ ~ c m ( 6 ) an < -aij - (cu + Cfj) ■ ]i n a U S CiiJJ-n a {] < -Cijjin (7) an < a^ + (en - cu) • p lt i,j = 1, 2, then the MAS (3) is stable. Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 427 Because the adjacency matrix U of a graph has both positive and negative eigenvalues, the conditions (6) and (7) implicitly imply the assumption that the single system at the node is stable. In this way, as expected, we derive that it is not possible to stabilize a network of instable systems by acting only on the topological structure. Given a specified node dynamic, coupling strength and bound on the adjacency matrix U, by conditions (6) and (7) we can assess MAS stability. Moreover, the MAS robustness with respect to varying switching topology can be dealt by considering the span of the eigenvalue of the admissible structure topologies. As we will show in the follows, it is possible easily to evaluate the eigenvalues of Ag, given the eigenvalues of U in some simple and representative cases of interest. Corollary 2 Let the MAS composed of n identical MIMO system of order 1 and interconnected by the digraph G = (V,E,U), with adjacency matrix U with eigenvalues 1*1 < V-2 < • • • 1*n- If the node dynamic matrix A — a and the coupling matrix BC = c fulfill the conditions: a < -c-]i n if c> (8) a < -c-f*i if c < 0, (9) then the MAS (3) is stable. The Corollary 2 reduces the analytical result of Theorem 1 to the case of the consensus of integrator (R.O. Saber & R. Murray, 2004) with coupling gain c. Smaller c, higher is the degree of robustness of the network to the slower node dynamic. In the opposite, higher c reduces the stability margin of the MAS. Finally, for a fixed dynamic at the node, the maximum admissible coupling strength c depends on the maximum and minimum eigenvalues of the adjacency matrix: c < -— if c>0 (10) fin C > -— if C<0. (11) n Corollary 3 Let the MAS of n identical MIMO system of the m-th order, interconnected by the digraph G = (V,E, U), with adjacency matrix U with eigenvalues ]i\ < \ii < . . . ]i n . If the node dynamic matrix A = {fl;y} and the coupling matrix BC = {c«} are both upper or lower triangular matrix and fulfill the conditions: 0» < — c ii ' J*n if Cjj > (12) *« < ~ c ii ' Hi if c ii < 0/ ( 13 ) then the MAS (3) is stable. 428 Robust Control, Theory and Applications (a) (b) (c) Fig. 1. Procedure of redirectioning of links in a regular network (a) with increasing probability p. As p increases the network moves from regular (a) to random (c), becoming small world (b) for a critical value of p. n=20, k=4 Notice that if BC = c • l n , (10) and (11) become: c< c > mm \ai t i ]in min |flj, __z Ml If Ctf > If ca < (14) (15) and hence the stability of MAS is explicitly given as function of the network slowest node dynamic. Now we would like to point out the case of undirected topology with symmetric adjacency matrix 17. If we assume A and BC being symmetric, then Ag is symmetric with real eigenvalue. Moreover from the field value property (Horn R.A. & Johnson C.R., 1995), let cr{A) = {&;} and a(BC) = {vj} the eigenvalues set of A and BC, then the eigenvalues of A + ]i\BC are in the interval [miny{#y} + f/ z miny{vy}, maxy{ay} + /^-maxy{vy}], for every l<i<n,l<j<m. In this way, there is a bound need to be satisfied by the topology structure, node dynamic and coupling matrix for MAS stabilization. Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 429 In the literature, the MAS consensuability results have been given in terms of Laplacian matrix properties. Here, differently, we have given bounds as function of the adjacency matrix features. Anyway we can use the results on the Laplacian eigenvalue for recasting the bounds given on the adjacency matrix. To this aim, defined the degree d{ of i-th node of an undirected graph as £/ uu, the Laplacian matrix is defined as L = D — U with D is the diagonal matrix with the degree of node i-th in position i-th. Clearly L is a zero row sums matrix with non-positive off-diagonal elements. It has at least one zero eigenvalue and all nonzero eigenvalues have nonnegative real parts. So U = D — L and being the minimum and maximum Laplacian eigenvalues respectively bounded by and the highest node degree, we have: Lemma 2 Let U the adjacency matrix of undirected and connected graph G = (V,E,U), with eigenvalues ]i\ < \ii < . . . < ]i n , then results: ^i(LT) > mindj — min(max{^ + d; : (k,j) <G E(G)},n) (16) i k,j ]i n {U) < maxdi (17) i Proof Easily follows from the Laplacian eigenvalues bound and the field value property (Horn R.A. & Johnson C.R., 1995). 4. Simulation validation In the follows we will present a variety of simulations to validate the above theoretical results under different kinds of node dynamic and network topology variations. Specifically the MAS topology variations have been carried out by using the well known Watts-Strogats procedure described in (Watts & S. H. Strogatz, 1998). In particular, starting from the regular network topology (p = 0), by increasing the probability p of rewiring the links, it is possible smoothly to change its topology into a random one (p = 1), with small world typically occurring at some intermediate value. In so doing neither the number of nodes nor the overall number of edges is changed. In Fig. 1 it shown the results in the case of MAS of 20 nodes with each one having k = 4 neighbors. Among the simulation results we focus our attention on the maximum and minimum eigenvalues of the matrixes U (i.e. ]i n and ]i{) and A g (i.e. Am an d A m ) and their bounds computed by using the results of the previous section. In particular, by Lemma 2, we convey the bounds on U eigenvalues in bounds on Ag eigenvalues suitable for the case of time varying topology structure. We assume in the simulations the matrices A and BC to be symmetric. In this way, if U eigenvalues are in \v\, v^\, let cr(A) = {a z }, a(BC) = {v z }, the eigenvalues of Ag will be in the interval [min oci + min^yy, vi v ; -}, max oi{ + maxjz^vy, v^ v ; -}] for i,j = 1, 2, . . . , n. i j J J i J Notice that, known the interval of variation \v\, v^\ of the eigenvalues set of U under switching topologies, we can recast the conditions (8), (9), (12), (13), (6), (7) and to use it for design purpose. Specifically, given the interval \o\, V2] associated to the topology possible variations, we derive conditions on A or BC for MAS consensuability. We consider a graph of n = 400 and k = 4. In the evolving network simulations, we started with k = 4 and bounded it to the order of O(log(n)) for setting a sparse graph. In Tab 1 are drawn the node dynamic and coupling matrices considered in the first set of simulations. 430 Robust Control, Theory and Applications -2 10 , 1 1 ■ 10"' (a) 10" 10 u (b) -10 -15 -20 10" • i i u r i« I * i* i 10"' 10" 10" -2 -4 -6 - -8 -10 -12 -14 -16 10 .^M— - "V" 'A 10"' 10" 10" (c) (d) Fig. 2. Case 1. Dashed line: bound on the eigenvalue; continuous line: eigenvalues, (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of LI 20 15 20 40 60 80 100 t Fig. 3. Case 1: State dynamic evolution in the time Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 431 (a) (b) -2 - -4 i 6 i v » _*; •*'%« -^ .,_ -8 * .»- 1 *V ! , =5. v :::V»u. -10 -12 . t -14 10 ' 10 10 10 (c) (d) Fig. 4. Case 2. Dashed line: bound on the eigenvalue; continuous line: eigenvalue: (a) Maximum eigenvalue of Ag, (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g , (d) Minimum eigenvalue of U A B C Case 1: -4.1 1 1 Case 2: -12 1 1 Case 3: -6 1 1 Case 4: -6 2 1 Table 1. Node system matrices (A,B,C) 432 Robust Control, Theory and Applications (a) (b) -10 E -15 -20 10"* • in • 10"' (c) 10 10 u 10" v — ^ > LI -5 1 ■ " " " V" **' " :-'» : =r -10 -15 ::::::::* >A w\ 4 i* ft* 10 10" p (d) 10° Fig. 5. Case 3. Dashed line: bound on the eigenvalue; continuous line: eigenvalue: (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of U 8 10 Fig. 6. Case 3: state dynamic evolution in the time Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 433 m 10"' 10" 10 u if 10"' 10" 10" (a) (b) -10 -15 -20 ^ E -25 -30 -35 -40 10 - W I,, 10"' 10" 10 u U t (c) (d) Fig. 7. Case 4. Dashed line: bound on the eigenvalues; continuous line: eigenvalues: (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of U In the case 1 (Fig 2), we note as although we start from a stable MAS network, the topology variation leads the network instability condition (namely Am becomes positive). In Fig. 3 it is shown the time state evolution of the firsts 10 nodes, under the switching frequency of 1 Hz. We note as the MAS converges to the consensus state till it is stable, then goes in instability condition. In the case 2, we consider a node dynamic faster than the maximum network degree dyi of all evolving network topologies from compete to random graph. Notice that although this assures MAS consensuability as drawn in Fig. 4, it can be much conservative. In the case 3 (Fig 5), we consider a slower node dynamic than the cases 2. The MAS is robust stable under topology variations. In Fig. 6 the state dynamic evolution is convergent and the settling time is about 4.6/ 1 Am (^g)l- Then we have varied the value for BC by doubling the B matrix value leaving unchanged the node dynamic matrix. As appears in Fig. 7, the MAS goes in instability condition pointing out 434 Robust Control, Theory and Applications »•'?» 1 . ■ p* f -» -;»ni I 10 1 V ll'l 8 J«#l| , ::; J 5 • .v -ir "% b :: S ; 4 :l:::::: . »» » ( ■»- 10"' 10" 10 u 10"' 10"' 10" 10" (a) (b) -10 -15 -20 -25 -30 10 V I 10 10" p (c) 10 u -10 -15 10" v r 'i\ >\ i 10 10" p (d) 1(T Fig. 8. Case 5. Dashed line: bound on the eigenvalue; continuous line: eigenvalue: (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of U that also the coupling strength can affect the stability (as stated by the conditions (8), (9)) and that this effect can be amplified by the network topological variations. A B C Case 5: -6 3 3 -12 "1" [10] Case 6: -3 3 3 -6 1 [10] Case 7: -3 3 3 -6 0.25 [10] Table 2. Node system matrices (A,B,C). Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 435 4 3 2" 1 L 10"' 10"' 10" 10° / - 10"' 10" 10" (a) (b) -8 -10 -12 .- - :l ~ -14 I v '*v. -16 i. ;;;;;;;- -18 -20 "l ::•::: - ^ ! 1 -2 -4 -6 -8 -10 -12 10"* 10"' 10" 10 u -14 10 if*'* 10"' 10" 10 u (c) (d) Fig. 9. Case 6. Dashed line: bound on the eigenvalue; continuous line: eigenvalue: (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of U On the other side, a reduction on BC increases the MAS stability margin. So we can tune the BC value in order to guarantee stability or desired robust stability MAS margin under a specified node dynamic and topology network variations. Indeed if BC has eigenvalues above 1, its effect is to amplify the eigenvalues of U and we need a faster node dynamic for assessing MAS stability. If BC has eigenvalues less of 1, its effect is of attenuation and the node dynamic can be slower without affecting the network stability. Now we consider SISO system of second order at the node as shown in Tab.2. In this case the matrix BC has one zero eigenvalue being the rows linearly dependent. In the case 5 the eigenvalues of A are oc\ = —4.76 and #2 = —13.23, the eigenvalues of the coupling matrix BC are V\ = 1 and v^ = 0. In this case the node dynamic is sufficiently fast for guaranteeing MAS consensuability (Fig. 8). In the case 6, we reduce the node dynamic matrix A to oci = — 1.15 e #2 — —7.85. Fig. 9 shows instability condition for the MAS network. We 436 Robust Control, Theory and Applications 1 n- 0.5 -0.5 • it i' " ' -'-'-'-- - '- i i*» ? _• i* ti «i V 10"^ 10"' 10" 10° ._* ji ii ii 10"' 10" 10" (a) (b) 1 1 I *% -2 -4 -6 -8 -10 -12 -14 10 :'•>": V^ 5^. m 10"' 10" 10 u (c) (d) Fig. 10. Case 7. Dashed line: bound on the eigenvalue; continuous line: eigenvalue, (a) Maximum eigenvalue of A g/ (b) Maximum eigenvalue U, (c) Minimum eigenvalue of A g/ (d) Minimum eigenvalue of U can lead the MAS in stability condition by designing the coupling matrix BC as appear by the case 7 and the associate Fig. 10. 4.1 Robustness to node fault Now we deal with the case of node fault. We can state the following Theorem. Theorem 2 Let A and BC symmetric matrix and G(V, E, U) an undirected graph. If the MAS system described by A g is stable, it is stable also in the presence of node faults. Moreover the MAS dynamic becomes faster after the node fault. Proof Being the graph undirected and A and BC symmetric then Ag is symmetric. Let Ag the MAS dynamic matrix associated to the network after a node fault. Ag is obtained from Ag by eliminating the rows and columns corresponding to the nodes went down. So Ag is a minor of Ag and for the interlacing theorem (Horn R.A. & Johnson C.R., 1995) it has eigenvalues inside Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 437 (a) (b) :£" "3 -3.5 (c) (d) Fig. 11. Eigenvalues in the case 1 = 1. Dashed line: eigenvalue in the case of complete topology with n = 100; continuous line: eigenvalue in the case of node fault: (a) Maximum eigenvalue of A g , (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g , (d) Minimum eigenvalue of U the real interval with extremes the minimum and maximum Ag eigenvalues. Hence if Ag is stable, Ag is stable too. Moreover, the maximum eigenvalue of Ag is less than one of Ag. So the slowest dynamic of the system x(t) = Agx(t) is faster than the system x(t) = Agx(t). In the follows we will show the eigenvalues of MAS dynamic in the presence of node fault. We consider MAS network with n = 100. We compare for each evolving network topology at each time simulation step, the maximum and minimum eigenvalues of Ag than those ones resulting with the fault of randomly chosen / nodes. Figures 11 and 12 show the eigenvalues of system dynamic for the cases / = 1 and I = 50. Notice that as the eigenvalues of U and Ag of fault network are inside the real interval containing the eigenvalues of U and Ag of the complete graph. In Fig. 13 are shown the time evolutions of state of the complete and faulted graphs. Notice that the fault network is faster than the initial network as stated by the analysis of the spectra of A g and A g . 438 Robust Control, Theory and Applications (a) (b) -1.5 -2 -2.5 t. . -3 -3.5 -4 10" v (c) 10 10" p (d) 1(T Fig. 12. Eigenvalues in the case of / = 50. Dashed line: eigenvalue in the case of complete topology with n = 100; continuous line: eigenvalue in the case of node fault: (a) Maximum eigenvalue of A g , (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g , (d) Minimum eigenvalue of U 5. Conclusions In this book chapter we have investigated the consensuability of the MASs under both the dynamic agent structure and communication topology variations. Specifically, it has given consensusability conditions of linear MASs as function of the agent dynamic structure, communication topology and coupling strength parameters. The theoretical results are given by transferring the consensusability problem to the stability analysis of LTI-MASs. Moreover, it is shown that the interplay among consensusability, node dynamic and topology must be taken into account for MASs stabilization: consensuability of MASs is assessed for all topologies, dynamic and coupling strength satisfying a pre-specified bound. From the practical point of view the consensuability conditions can be used for both the analysis and planning of MASs protocols to guarantee robust stability for a wide range of possible interconnection topologies, coupling strength and node dynamics. Also, the consensuability Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics 439 100 100 50 X -50 I ife^ Fig. 13. Time evolution of the state variables for 1=50: top Figure: complete graph. Bottom Figure: graph with fault. of MAS in the presence of node faults has been analyzed. Simulation scenarios are given to validate the theoretical results. Acknowledgement The author would like to thank Ms. F. Schioppa for valuable discussion. 6. References J.K. Hedrick, D.H. McMahon, V.K. Narendran, and D. Swaroop. (1990). Longitudinal vehical controller design for WHS systems. Proceedings of the American Control Conference, pages 3107-3112. P. Kundur. 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Topology independent controller design for networked systems, IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, Dicembre 2004 D. J. Watts, S. H. Strogatz, (1998). Collective dynamics of small world networks, Nature - Macmillan Publishers Ltd, Vol. 393, Giugno 1998. Horn R.A. and Johnson C.R., (1995). Topics in Matrix Analysis Cambridge University Press 1995. 19 On Stabilizability and Detectability of Variational Control Systems Bogdan Sasu*and Adina Lumini^a Sasu Department of Mathematics, Faculty of Mathematics and Computer Science, West University ofTimi§oara, V. Pdrvan Blvd. No. 4 300223 Timi§oara Romania 1. Introduction The aim of this chapter is to present several interesting connections between the input-output stability properties and the stabilizability and detectability of variational control systems, proposing a new perspective concerning the interference of the interpolation methods in control theory and extending the applicability area of the input-output methods in the stability theory Indeed, let X be a Banach space, let (0, d) be a locally compact metric space and let S = X x 0. We denote by B(X) the Banach algebra of all bounded linear operators on X. If Y, U are two Banach spaces, we denote by B(U, Y) the space of all bounded linear operators from U into Y and by C s (0, B(U,Y)) the space of all continuous bounded mappings H : — > B(U,Y). With respect to the norm 1 1 \H\ \ \ := sup | |H(0) 1 |, C s (0, B(U, Y)) is a Banach space. If H e C S (Q,B(U,Y)) and Q e C S (Q,B(Y,Z)) we denote by QH the mapping 3 0^ Q(6)H(6). It is obvious that QH e C S (Q,B(U,Z)) . Definition 1.1. Let / 6 {IR + ,]R}. A continuous mapping a : x / — > is called a flow on if a(6, 0) = 6 and a(6, s + t) = a{a{6, s), t), for all (6, s,t) e x J 2 . Definition 1.2. A pair n = (^c) is called a linear skew-product flow on 6 = X x if a is a flow on and O:0x IR + — > 23 (X) satisfies the following conditions: (i) <&(0, 0) = J d/ the identity operator on X, for all e 0; (ii)<S>(0,* + s) = &(<r(0,t),s)®(6,t), for all (0,*,s) G x IR 2 . (the cocycle identity); (iii) (0, f ) h^ O(0, £)x is continuous, for every x £ X; (iv) there are M > 1 and cv > such that | |O(0, *) 1 1 < Me^, for all (0, t) G x R + . The mapping is called f/ze cocycle associated to the linear skew-product flow tt = (<E>,c). Let L / 1 oc (IR + , X) denote the linear space of all locally Bochner integrable functions u : R+ -^ X. Let 7i = (0,(j) be a linear skew-product flow on S = X x 0. We consider the variational integral system Jo (S n ) x e (t;xQ,u) = ®(0,t)x o + ®(a(6,s),t-s)u(s) ds, t>0,6e® * The work is supported by The National Research Council CNCSIS-UEFISCSU, PN II Research Grant ID 1081 code 550. 442 Robust Control, Theory and Applications with u G L Z 1 0C (R+,X) and x G x - Definition 1.3. The system (S n ) is said to be uniformly exponentially stable if there are N, v > such that IM*;*o/0)|| < Ne- Vt \\x \\, V(0,*) g x R + ,Vx e x. Remark 1.4. It is easily seen that the system (S^) is uniformly exponentially stable if and only if there are N,v > such that ||<S>(0,*)|| < Ne~ vt , for all (0, t ) G © x R+. If 7i = (0,(7) is a linear skew-product flow on S = X x and P G C s (0, £>(X)), then there exists a unique linear skew-product flow denoted zip = (<&p,o~) on X x such that this satisfies the variation of constants formula: ® P (e,t)x = ®(e,t)x+ [ ®(o-(6,s),t-s)PM6,s))®p(6,s)xds (1.1) Jo and respectively ® P (0,t)x = ®(0,t)x+ [ <3>p(cr(6,s),t-s)PM6,s))®(e,s)xds (1.2) Jo for all (x,0,t) G £ x R+. Moreover, if M,o; are the exponential growth constants given by Definition 1.2 (iv) for n, then ||<D P (6U)|| <M^ a;+M ll p ll)^ V(0,f) G0xR + . The perturbed linear skew-product flow zip = (Op, a) is obtained inductively (see Theorem 2.1 in (Megan et al., 2002)) via the formula 00 n=0 where Jo ®o(e,t)x = ®(e,t)x and ® n (6,t)x= / ®(cr(e,s),t - s) P((r(0,s)) ® n _ 1 (6,s)xds,n > 1 for every (x, 0) G S and £ > 0. Let 17, Y be two Banach spaces, let B G C S (0,H(17,X)) and C G C s (0,£(X, Y)). We consider the variational control system (n, B, C) described by the following integral model (x(6,t,x ,u) = ®(e,t)x + fQ®(cr(0,s),t - s)B(a(6,s))u(s)ds y(6,t,x ,u) = C(cr(6,t))x(6,t,x ,u) where t > 0, (x ,6) G S and u G LJ- 0C (R +/ 17). Two fundamental concepts related to the asymptotic behavior of the associated perturbed systems (see (Clark et al, 2000), (Curtain & Zwart, 1995), (Sasu & Sasu, 2004)) are described by stabilizability and detectability as follows: Definition 1.5. The system [re, B, C) is said to be: (i) stabilizable if there exists a mapping F G C s (0,B(X,U)) such that the system (S nBF ) is uniformly exponentially stable; (ii) detectable if there exists a mapping K G C s (0,B(Y,X)) such that the system (S nKC ) is uniformly exponentially stable. On Stabilizability and Detectability of Variational Control Systems 443 Remark 1.6. (i) The system (n, B, C) is stabilizable if and only if there exists a mapping F £ C s (0, B (X, IT)) and two constants N, v > such that the perturbed linear skew-product flow n BF — {^ > B¥/ cr ) nas the property \\®bf(0, t)\\< Ne~ vt , V(0, t) e x R+; (ii) The system (n,B,C) is detectable if and only if there exists a mapping K G C s (0, £>(Y, X)) and two constants N, v > such that the perturbed linear skew-product flow n K c = (^ko cr ) has the property \\®kc(0, 1 1 < N*" 1 */ V(0, G x R+. In the present work we will investigate the connections between the stabilizability and the detectability of the variational control system (n,B,C) and the asymptotic properties of the variational integral system (S n ). We propose a new method based on input-output techniques and on the behavior of some associated operators between certain function spaces. We will present a distinct approach concerning the stabilizability and detectability problems for variational control systems, compared with those in the existent literature, working with several representative classes of translations invariant function spaces (see Section 2 in (Sasu, 2008) and also (Bennet & Sharpley, 1988)) and thus we extend the applicability area, providing new perspectives concerning this framework. A special application of our main results will be the study of the connections between the exponential stability and the stabilizability and detectability of nonautonomous control systems in infinite dimensional spaces. The nonautonomous case treated in this chapter will include as consequences many interesting situations among which we mention the results obtained by Clark, Latushkin, Montgomery-Smith and Randolph (see (Clark et al., 2000)) and the authors (see (Sasu & Sasu, 2004)) concerning the connections between stabilizability, detectability and exponential stability. 2. Preliminaries on Banach function spaces and auxiliary results In what follows we recall several fundamental properties of Banach function spaces and we introduce the main tools of our investigation. Indeed, let M (R+, R) be the linear space of all Lebesgue measurable functions u : R+ — > IR, identifying the functions equal a.e. Definition 2.1. A linear subspace B of M (IR+, IR) is called a normed function space, if there is a mapping | • |g : B —> R+ such that: (z) \u\b = if and only if u = a.e.; (ii) \ocu\b = \oc\ \u\b, for all (oc,u) e R x B; (Hi) \u + v\b < \u\b + \v\b, for all u,v € B; (iv) if \u(t)\ < \v(t)\ a.e. t e R+ and v G B, then u e Band \u\ B < \v\ B . If (B, | • |g) is complete, then B is called a Banach function space. Remark 2.2. If (B, | • |g) is a Banach function space and u G B then \u(-)\ G B. A remarkable class of Banach function spaces is represented by the translations invariant spaces. These spaces have a special role in the study of the asymptotic properties of the dynamical systems using control type techniques (see Sasu (2008), Sasu & Sasu (2004)). 444 Robust Control, Theory and Applications Definition 2.3. A Banach function space (B, | • |b) is said to be invariant to translations if for every u : R+ — > IR and every t > 0, w G B if and only if the function to to z' \ ( U(s-t) ,S>t belongs to B and \ut\$ = \u\b- Let C C (R+,R) denote the linear space of all continuous functions v : R+ — > R with compact support contained in R + and let L* (R+,R) denote the linear space of all locally integrable functions u : R+ — > R. We denote by T(R+) the class of all Banach function spaces B which are invariant to translations and satisfy the following properties: (i)C c (R + ,R)cBcL | 1 oc (R+,R); (ii) if B \ L 1 (R + , R) ^ then there is a continuous function ^B\L ! (R +/ R). For every A C R+ we denote by xa the characteristic function of the set A. Remark 2.4. (i) If B G T(R+), then X[o,t) € B > for a11 t > °- (ii) Let B G T(R+), weB and t > 0. Then, the function u t : R+ -> R, fl f (s) = w(s + t) belongs to B and \ti t \ B < \u\ B (see (Sasu, 2008), Lemma 5.4). Definition 2.5. (i) Let u,v G A4(R+,R). We say that w and i? are equimeasurable if for every t > the sets {s G R+ : |w(s) | > t} and {s G R+ : |u(s) | > t} have the same measure, (ii) A Banach function space (B, \ • |b) is rearrangement invariant if for every equimeasurable functions w,i? : R+ — > R+ with w G B we have that »G6 and |w|g = |i?|g. We denote by 7^(R+) the class of all Banach function spaces B G T(R+) which are rearrangement invariant. A remarkable class of rearrangement invariant function spaces is represented by the so-called Orlicz spaces which are introduced in the following remark: Remark 2.6. Let cp : R+ — ► R+ be a non-decreasing left-continuous function, which is not identically zero on (0, oo). The Young function associated with cp is defined by Y(p(t) = Jo <p(s) ds. For every u G A^(R+,R) let M<p(u) := f™ Y<p{\u{s)\) ds. The set O^ of all u G A^(R+,R) with the property that there is k > such that Mq,(ku) < oo, is a linear space. With respect to the norm \u\q, := mi{k > : Mq,(u/k) < 1}, O^ is a Banach space, called the Orlicz space associated with cp. The Orlicz spaces are rearrangement invariant (see (Bennet & Sharpley, 1988), Theorem 8.9). Moreover, it is well known that, for every p G [1, oo], the space L^(R + , R) is a particular case of Orlicz space. Let now (X, || • ||) be a real or complex Banach space. For every B G T(R+) we denote by B(R+,X), the linear space of all Bochner measurable functions u : R+ — ► X with the property that the mapping N u : R+ — > R+, N u (t) = \\u(t)\\ lies in B. Endowed with the norm IMIb(jr +/ x) : = |N W | B ,B(R + ,X) is a Banach space. Let (0,d) be a metric space and let S = X x 0. Let n = (0,(7") be a linear skew-product flow on 8 = X x 0. We consider the variational integral system (S n ) x (t;xo,u) = ®(O,t)xo+ [ ®(cr(6,s),t-s)u(s)ds, £>O,0G0 Jo On Stabilizability and Detectability of Variational Control Systems 445 with u G L Z 1 0C (R +/ X) and x € x - An important stability concept related with the asymptotic behavior of dynamical systems is described by the following concept: Definition 2.7. Let W G T(R+). The system (S n ) is said to be completely (W(R+,X), W(R+, X))-stable if the following assertions hold: (i) for every u G W(R+, X) and every G the solution x (• ; 0, u) G W(R+, X); (ii) there is A > such that ||x0(-;O,tt)|| W ( R+X ) < A||w|| W ( R+X ), for all (u,9) G W(R+,X) x 0. A characterization of uniform exponential stability of variational systems in terms of the complete stability of a pair of function spaces has been obtained in (Sasu, 2008) (see Corollary 3.19) and this is given by: Theorem 2.8. Let W G 7£(R+). The system (S n ) is uniformly exponentially stable if and only if (S n ) is completely (W(R+,X), W(R+,X)) -stable. The problem can be also treated in the setting of the continuous functions. Indeed, let Q>(R+,R) be the space of all bounded continuous functions u : R+ —> R. Let Cq(R+,R) be the space of all continuous functions u : R+ — > R with lim u(t) = and let Cqo(R+,R) := {u G C (R+,R) : u(0) = 0}. Definition 2.9. Let V G {C fo (R + ,R),C (R+,R),Coo(R+,R)}. The system (S n ) is said to be completely (V(R+, X), V r (R+ / X))-stable if the following assertions hold: (i) for every w G V r (R+,X) and every 6 G the solution xg(- ;Q,u) G y(R +/ X); (ii)thereisA > such that ||^(-;0,w)|| y(]R+/X) < A||w||y (]R+/X ),forall (u,6) G V(R+,X) x 0. For the proof of the next result we refer to Corollary 3.24 in (Sasu, 2008) or, alternatively, to Theorem 5.1 in (Megan et al, 2005). Theorem 2.10. Let V G {Q(R + ,R),C (R + ,R),C o(IR+,R)}. The system (S n ) is uniformly exponentially stable if and only if (S n ) is completely (V(R+, X), V(R+, X))-stable. Remark 2.11. Let W G 7e(R+) U {C (R+,X),C o(R+,X),C & (R + ,X)}. If the system (S n ) is uniformly exponentially stable then for every 6 G the linear operator P e w : W(R+,X) -+ W(R+,X), (P e w u)(t)= / <&(cr(e,s),t - s)u(s) ds is correctly defined and bounded. Moreover, if A > is given by Definition 2.7 or respectively by Definition 2.9, then we have that sup 0G0 | |PJy| | < A. These results have several interesting applications in control theory among we mention those concerning the robustness problems (see (Sasu, 2008)) which lead to an inedit estimation of the lower bound of the stability radius, as well as to the study of the connections between stability and stabilizability and detectability of associated control systems, as we will see in what follows. It worth mentioning that these aspects were studied for the very first time for the case of systems associated to evolution operators in (Clark et al., 2000) and were extended for linear skew-product flows in (Megan et al., 2002). 446 Robust Control, Theory and Applications 3. Stabilizability and detectability of variational control systems As stated from the very beginning, in this section our attention will focus on the connections between stabilizability, detectability and the uniform exponential stability Let X be a Banach space, let (0,d) be a metric space and let n = (0,cr) be a linear skew-product flow on 8 = X x 0. We consider the variational integral system (Sn) x e (t;x ,u) =®(e,t)x + [ ®M6,s),t-s)u(s)ds, £>O,0G0 Jo with u G L / 1 0C (R+,X) and x G X. Let U, Y be Banach spaces and let B G C s (0, B(U, X)), C G C s (0, B(X, Y)). We consider the variational control system (n,B,C) described by the following integral model ( x(6, t, x , u) = <S>(0, t)x + f* ®((r(6, s), t - s)B(cr(e, s))u(s) ds [y(0,t,x o ,u) = C(or(0,t))x(9,t,x o ,u) where t > 0, (x ,6) e £ and u G ^/ 1 0C (IR+, U). According to Definition 1.5 it is obvious that if the system (S n ) is uniformly exponentially stable, then the control system (n,B,C) is stabilizable (via the trivial feedback F = 0) and this is also detectable (via the trivial feedback K = 0). The natural question arises whether the converse implication holds. Example 3.1. Let X = R,0 = IR and let <r{Q,t) = + t. Let (S n ) be a variational integral system such that <b(Q,t) = I d (the identity operator on X), for all (0,t) G x R + . Let U = Y = X and let B(Q) = C(0) = I d , for all 6 G 0. Let 5 > 0. By considering F(0) = -^ I d , for all G 0, from relation (1.1), we obtain that Jo ® BF (6,t)x = x-5 I ® BF (6,s)xds, Vf>0 for every (x,6) G S. This implies that ® BF (6,t)x = e~ st x, for alH > and all (x,6) G S, so the perturbed system (S nBF ) is uniformly exponentially stable. This shows that the system (n,B,C) is stabilizable. Similarly, if 8 > 0, for K(9) = —S Id, for all G 0, we deduce that the variational control system [n, B, C) is also detectable. In conclusion, the variational control system (n,B,C) is both stabilizable and detectable, but for all that, the variational integral system (S n ) is not uniformly exponentially stable. It follows that the stabilizability or /and the detectability of the control system (n,B,C) are not sufficient conditions for the uniform exponential stability of the system (S n ). Naturally, additional hypotheses are required. In what follows we shall prove that certain input-output conditions assure a complete resolution to this problem. The answer will be given employing new methods based on function spaces techniques. Indeed, for every 6 G 0, we define P e : 4 C (R+,X) -+ 4 C (R+,X), (P e w)(t) = J*®((r(e,s),t-s)w(s) ds On Stabilizability and Detectability of Variational Control Systems 447 and respectively B e : LJnCR+,11) - 4c( K +- x )' (B 9 ")( f ) = B(a(0,t))u(t) C e : 4, C (R+,X) - LJ 0C (R + ,Y), (C e v)(t) = C(a(9,t))v(t). We also associate with the control system S = (n,B,C) three families of input-output mappings, as follows: the left input-output operators {L e } ee ® defined by L e : LJJR+, U) -> L} 0C (TR + , X), L e := P B the right input-output operators {R e }ee© given by R e : L | 1 0C (R + , X) - L | 1 0C (R + , Y), £ := C^P and respectively the global input-output operators {G^}^© defined by G : L^ 0C (R+, LI) -> 4, C (R +/ Y), G := C^P^B . A fundamental stability concept for families of linear operators is given by the following: Definition 3.2. Let Z\, Z 2 be two Banach spaces and let W £ T(R+) be a Banach function space. A family of linear operators {O e : L | 1 oc (IR + ,Z 1 ) — > 1^(11+, Z 2 )}0 G0 is said to be (W(R+, Zi), W(R + , Z2)) -stable if the following conditions are satisfied: (i) for every oc x £ WCR+rZ^ and every 6 £ 0, 0% £ W(R+,Z 2 ); (n) there ism > such that | |O ^i 1 1 W ( ]R+ Zz ) < w lkillw(R+ z x )> for a11 a i G W(R+,Z!) and all e 0. Thus, we observe that if W € 7£(R+), then the variational integral system (S n ) is uniformly exponentially stable if and only if the family {P e }ee© is (W(R+,X), W(R+,X)) -stable (see also Remark 2.11). Remark 3.3. Let Z\, Zi be two Banach spaces and let W £ T(R+) be a Banach function space. If Q £ C s (0, B(Zi, Z 2 )) then the family {Q% G0 defined by Q e : 4 C (R +/ Zi) - 4 C (R+,Z 2 ), (Q*«)(0 = Q(^, *))*(') is (W(R+,Zi), W(R+,Z 2 )) -stable. Indeed, this follows from Definition 2.1 (iv) by observing that ll(Q**)(OII < IIIQIII IWOII, Vf > o,v* g w(R+,z 1 ),V0 £ 0. The main result of this section is: Theorem 3.4. Let Wbea Banach function space such that W £ 7£(R+). The following assertions are equivalent: (i) the variational integral system (S n ) is uniformly exponentially stable; (ii) the variational control system (n,B,C) is stabilizable and the family of the left input-output operators {L e } 9e@ is (W(R+ / U r ) / W r (R+ / X))-stefck; (Hi) the variational control system (n,B,C) is detectable and the family of the right input-output operators {R e } ee © is (W r (R +/ X) / W(R+ / Y))-steJbk (iv) the variational control system (tc,B,C) is stabilizable, detectable and the family of the global input-output operators {G e } 9e@ is (W(R + ,L7), W(R+, Y))-stable. 448 Robust Control, Theory and Applications Proof. We will independently prove each equivalence (i) <^=> (ii), (i) -<=> (Hi) and respectively (z) «=>• (iv). Indeed, we start with the first one and we prove that (i) => (ii). Taking into account that (S n ) is uniformly exponentially stable, we have that the family {P 6 heG is (W(R+,X), W(R+,X))-stable. In addition, observing that ||(L^)(0|| < sup ||P || HIBIII HiiWIl V« G W(R+,U),V0 G ee© from Definition 2.1 (iv) we deduce that that the family {L e }o e @ is (W(R+,lf),W(R+,X))-stable. To prove the implication (ii) => (i), let F G C S (S / B(X / U)) be such that the system (S nBF ) is uniformly exponentially stable. It follows that the family {H e }Q E @ is ( W(R+, X), W(R+, X) ) -stable, where H e : L} 0C (R+,X) -► L | 1 0C (R + ,X), (H e u)(t) = / <b BT (p-{p,s),t-s)u(s)ds, t > 0,6 G 0. For every G let F : 4 C (R + ,X) - 4, C (R+,U), (F*u)(f) = F(cr(0,O)u(*). Then from Remark 3.3 we have that the family {F e } ee@ is (W(R+, X), W(R+, U)) -stable. Let G and let w G L | 1 oc (R + , X). Using Fubini's theorem and formula (1.1), we successively deduce that (L e F°H e u)(t)= f f ®M6,s),t-s)B(cr(6 / s))F(a(6,s))® BF (cr(e,T),s-T)u(T)dTds = Jo Jo = f f ®(a(e,s),t - s)B(a(6,s))F(a(6,s))® BF (a(6,T),s - t)u(t) dsdr = T ^(a(0 / T + ^J-T-^B(a(0 / T + ^)F(a(0 / T + ^)^ BF (a(0 / T) / ^u(T)d^dT: If Jo Jo = f [® BF (cr(6,T),t-T)u(T)-®(0-(6,T),t-T)u(T)]dT = = (H e u)(t)-(P e u)(t), W>0. This shows that P e u = H e u - L Q F Q H Q u, \/u G L} 0C (K + ,X),\/6 G 0. (3.1) Let m\ and m^_ be two constants given by Definition 3.2 (ii) for {H^}^ G @ and for {L^}^ G @, respectively. From relation (3.1) we deduce that P e u G W(R+,X), for every u G W(R+,X) and \\P 6 u\\w(K + ,x) < m 1 (l + m 2 |||F|||) \\u\\ w{K+/X) , Vu G W(R + ,X),V0 G 0. From the above relation we obtain that the family {P e }ee& is (W(R+, X), W(R+, X))-stable, so the system (S^) is uniformly exponentially stable. The implication (i) => (Hi) follows using similar arguments with those used in the proof of (i) => (ii). To prove (Hi) =^ (i), let K G C 8 (0,B(Y,X)) be such that the system (S nKC ) is uniformly exponentially stable. Then, the family {T e } ee@ is ( W(R + , X), W(R+, X))-stable, where T : 4 C (R+,X) - 4 C (R+,X), (1**0(0 = J*<P KC (cr(e,s),t-s)u(s)ds. On Stabilizability and Detectability of Variational Control Systems 449 For every 0G0 we define K e : L} 0C (R + ,Y) -+ 4 C (R + ,X), (K e u)(t) = K(a(Q f t))u(t). From Remark 3.3 we have that the family {K e } ee@ is (W(R + , Y), W(R+, X)) -stable. Using Fubini's theorem and the relation (1.2), by employing similar arguments with those from the proof of the implication (ii) =>• (i), we deduce that P e u = T e u - Y e K e R e u, Mu G L} 0C (R+,X),Ve G 0. (3.2) Denoting by <ft and by ^2 some constants given by Definition 3.2 (ii) for {T e }Q e ® and for {R e }ee©' respectively, from relation (3.2) we have that P e u G W(R+,X), for every u G W(R+,X)and ll^llw(R + ,X) < qi(l + q 2 \\\K\\\) |M| W(R+/X ), Vu € W(R + ,X),V0 G 0. Hence we deduce that the family {P e }ee© is (W(R+, X), W(R+, X))-stable, which shows that the system (S n ) is uniformly exponentially stable. The implication (i) => (iv) is obvious, taking into account the above items. To prove that (iv) =^ (/), let K G C s (0, B(Y, X)) be such that the system (S 7lKC ) is uniformly exponentially stable and let {K e }Q E @ and {r^}^ G be defined in the same manner like in the previous stage. Then, following the same steps as in the previous implications, we obtain that L e u = Y e B e u - T e K e G e u, \/u G L | 1 oc (R + ,X),V0 G 0. (3.3) From relation (3.3) we deduce that the family {L e } ee@ is (W(R+, U), W(R+,X))-stable. Taking into account that the system (n,B,C) is stabilizable and applying the implication (ii) =^ (i), we conclude that the system (S n ) is uniformly exponentially stable. □ Corollary 3.5. Let V G {Q(R + ,R),C (R+,R),C o(R+,R)}. The following assertions are equivalent: (i) the variational integral system (S n ) is uniformly exponentially stable; (ii) the variational control system (n r B,C) is stabilizable and the family of the left input-output operators {L e } 9e@ is (V(R+,U),V(R+,X))-stable; (Hi) the variational control system (tc,B,C) is detectable and the family of the right input-output operators {R e } ee ® is (V(R+,X),V(R+,Y))-stable (iv) the variational control system (n,B,C) is stabilizable, detectable and the family of the global input-output operators {G 6 }q e @ is (V(K +/ U),V(K +/ Y)) -stable. Proof. This follows using similar arguments and estimations with those from the proof of Theorem 3.4, by applying Theorem 2.10. □ 4. Applications to nonautonomous systems An interesting application of the main results from the previous section is to deduce necessary and sufficient conditions for uniform exponential stability of nonautonomous systems in terms of stabilizability and detectability. For the first time this topic was considered in (Clark et al., 2000)). We propose in what follows a new method for the resolution of this problem based on the application of the conclusions from the variational case, using arbitrary Banach function spaces. Let X be a Banach space and let !# denote the identity operator on X. 450 Robust Control, Theory and Applications Definition 4.1. A family U = {L7(f,s)}^> s >o C B(X) is called an evolution family if the following properties hold: (i) U(t,t) = I d andl7(£,s)l7(s,£ ) = LT(f,f ), for alH > s > t > 0; («) there are M > 1 and a; > such that 1 1 U(t, s) \ \ < Me w ^- S \ for all f > s > t > 0; (in) for every x G X the mapping (t,s) \-^ U(t,s)x is continuous. Remark 4.2. For every P G C S (R+, £>(X)) (see e.g. (Curtain & Zwart, 1995)) there is a unique evolution family Up = {Up(t,s)} t > s >o such that the variation of constants formulas hold: U P (t,s)x = U(t,s)x+ J U(t,T)P(T)U P (T,s)xdT, W>s >0,VxGX and respectively U P (t,s)x= U(t,s)x+ J U P (t,T)P{T)U{T,s)xdT, \/t >s >0,VxG X. Let U = {U{t, s) }t> s >Q be an evolution family on X. We consider the nonautonomous integral system (Sy) x s (t',XQ,u) = U(t,s)xQ+ I U(t / T)u(r) dr, t>s,s>0 with u G L / 1 0C (R+,X) and x G X. Definition 4.3. The system (Sy) is said to be uniformly exponentially stable if there are N, v > such that ||x s (*;x ,0)|| < Mr v ('- S )||;eo||,for all* > s > and all x G X. Remark 4.4. The system (Sy ) is uniformly exponentially stable if and only if there are N, v > such that ||lT(*,s)|| < Ne- V ^~ s \ for all* > s > 0. Definition 4.5. Let W G T(R+). The system (S w ) is said to be completely (W(R+,X), W(R+, X))-stable if for every w G W(R+,X), the solution x (-;0,w) G W(R+,X). Remark 4.6. If the system (S w ) is completely (W(R+,X), W(R+,X)) -stable, then it makes sense to consider the linear operator ? : W(R+,X) -► W(R+,X), ?{u) = x (')0,u). It is easy to see that IP is closed, so it is bounded. Let now U,Y be Banach spaces, let B G C S (R+, 23(17, X)) and let C G C S (R+,H(X, Y)). We consider the nonautonomous control system (U,B,C) described by the following integral model (x s (t;x ,u) = U(t,s)x + f*U(t,T)B(r)u(T) dz, t > s, s > y s (t;xQ,u) = C(t)x s (t;xQ,u), t > s, s > with u G Lj- 0C (R +/ LT),x G X. Definition 4.7. The system (£Y, B, C) is said to be: (i) stabilizable if there exists F G C S (R+,#(X, U)) such that the system (Sy BF ) is uniformly exponentially stable; (ii) detectable if there exists G G C S (R+,23(Y,X)) such that the system (Sy GC ) is uniformly exponentially stable. On Stabilizability and Detectability of Variational Control Systems 451 We consider the operators B : 4, C (R+,(I) - 4, C (]R + ,X), (Su)(f) = B(t)u(t) C : L^(R+,X) -> 4 C (R+,Y), (e«)(f) = B(f)«(0 and we associate with the system (U, B, C) three input-output operators: the left input-output operator defined by £ : Ll c (R+, U) -* LJJR+, X), L = TO the right input-output operator given by a : l} oc (r +i x) - l} oc (r +i y), % = &y and respectively the global input-output operator defined by S : LJ 0C (R+, U) -> 4, C (R+, Y), S = CTO. Definition 4.8. Let 7,\,Zi be two Banach spaces and let W G T(R+) be a Banach function space. An operator Q : L | 1 oc (IR + ,Z 1 ) —> L | 1 oc (R + ,Z2) is said to be (W(R+,Z 1 ),W(R+,Z 2 ))-stable\£ for every A G W(R+,Zi) the function Q A G W(R+,Z 2 ). The main result of this section is: Theorem 4.9. Let W be a Banach function space such that B G 7£(R+). The following assertions are equivalent: (i) the integral system (Sy) is uniformly exponentially stable; (ii) the control system (U,B,C) is stabilizable and the left input-output operator L is (W(R + ,U),W(R + ,X))-stable; (Hi) the control system (U,B,C) is detectable and the right input-output operator % is ( W(R+, X), W(R+, Y))-stable; (iv) the control system (U, B, C) is stabilizable, detectable and the global input-output operator S is (W(R+,U),W(R+,Y))-stable. Proof. We prove the equivalence (i) <^=> (ii) , the other equivalences: (i) -<=> (Hi) and (i) -<=> (iv) being similar. Indeed, the implication (z) =>• (ii) is immediate. To prove that (ii) => (i) let = R+, a : x R + -> ®,o-(Q,t) = 6 + t and let ®(0,t) = U(t + 9,6), for all (0,f) G x R + . Then 7i = (<3>, c) is a linear skew-product flow and it makes sense to associate with n the following integral system (S n ) x e (t;x ,u) =®(e,t)x + [ <&M6,s),t-s)u(s)ds, £>O,0G0 Jo with u G L | 1 oc (R + ,X) and x ^ x - We also consider the control system (n,B,C) given by !x(6,t,x ,u) = <&(e,t)xQ + f t Q <&(o-(e,s) / t-s)B(o-(6 / s))u(s)ds y(6,t,x ,u) = C(o-(6,t))x(6,t,x ,u) where t > 0, (xq,0) G £ and w G Lj- 0C (R+, IT). For every 6 G we associate with the system (zr, B, C) the operators P , B^ and L e using their definitions from Section 3. 452 Robust Control, Theory and Applications We prove that the family {L e } ee@ is (W(R + ,U), W(R +/ X)) -stable. Let 6 G and let a G W(R+, 17). Since W is invariant to translations the function belongs to W(R+, 17) and \M\w(iR +/ u) = \M\w(R+,u)- Since tne operator £ is (W(R+, 17), W(R+, X))-stable we obtain that the function <p:R+^X, ?(*) = (£**)(*) belongs to W(R+, X). Using Remark 2.4 (ii) we deduce that the function 7:R + ->X, j(t) = <p(t + Q) belongs to W(R+,X) and ||tIIw(r +/ x) < ll<pllw(R +/ x)- We observe that (L e oc)(t) = / U(6 + t,0 + s)B(6 + s)oc(s) ds = / U(6 + t f r)B(r)oc(r - 6) dr = r e+t = / U(e + t,r)B(r)oc e (r)dr=(Loc )(e + t)=y(t), \/t > 0. This implies that L^a: belongs to W(R+, X) and ll L ^llw(R +/ X) = ll7llw(R +/ X) < ll<pllw(R +/ X) < <ll^llll^llw(R + ,LI) = ll^lllkllw(R + ,L7)- (4-1) Since 6 G and a G W(R+, 17) were arbitrary from (4.1) we deduce that the family {L e }g e @ is (W(R + ,l7),W(R+,X))-stable. According to our hypothesis we have that the system (U, B, C) is stabilizable. Then there is F G C s (R+,i3(X, 17)) such that the (unique) evolution family U B f = {U B p(t,s)} t > s >o which satisfies the equation U BF (t,s)x = U(t,s)x+ J U{t,T)B{T)F{T)U BF (T,s)xdT, W>s>0,VxGX (4.2) has the property that there are N, v > such that ||LZ B f(^s)|| < Ne~ v ^- S \ \/t > s > 0. (4.3) For every (0,i) G x R+, let <b(6,t) := 17 B f(0 + £,0). Then, we have that 7t = (0>,<7-) is a linear skew-product flow. Moreover, using relation (4.2) we deduce that f ®(<r(6,s),t- s)BM6,s))F((r(6,s))&(6,s)x ds = Jo = f U(6 + t,0 + s)B(0 + s)F(0 + s)U BF (6 + s,6)xds = Jo r 6+t = / U(6 + t,T)B(T)F(T)U B¥ (T / e)xdT = J 6 On Stabilizability and Detectability of Variational Control Systems 453 = U BF (0 + t,0)x - U(0 + t,Q)x = <b(0,t)x - <$>(6,t)x (4.4) for all (6, t ) e x R + and all x e X. According to Theorem 2.1 in (Megan et al, 2002), from relation (4.4) it follows that O>(0, i) = O BF (0, t), V(0, t) e x R + so ft = ttbf- Hence from relation (4.3) we have that I|3>bf(M)II = \\U BF (6 + t,6)\\ <Ne~ vt , \/t > 0,V6> e which shows that the system (S^) is uniformly exponentially stable. So the system (n, B, C) is stabilizable. In this way we have proved that the system (S n ) is stabilizable and the associated left input-output family {L e } 9e@ is (W(R+, U), W(R+, X)) -stable. By applying Theorem 3.4 we deduce that the system (S n ) is uniformly exponentially stable. Then, there are N,5 > such that ||O(0/OII < Ne"^, Vf >O,V0G0. This implies that ||L7(f,s)|| = ||<D(s,*-s)|| <Ne- s ^~ s \ \/t > s > 0. (4.5) From inequality (4.5) and Remark 4.4 we obtain that the system (Sn) is uniformly exponentially stable. □ Remark 4.10. The version of the above result, for the case when W = I/(R + ,R) with p G [1, oo), was proved for the first time by Clark, Latushkin, Montgomery-Smith and Randolph in (Clark et al., 2000) employing evolution semigroup techniques. The method may be also extended for spaces of continuous functions, as the following result shows: Corollary 4.11. Let V e {C^(R + ,R),C (R+,R),C o(R+,R)}. The following assertions are equivalent: (i) the system (Sy) is uniformly exponentially stable; (ii) the system (U,B,C) is stabilizable and the left input-output operator £ is (V(R+,U),V(R+,X))-stable; (Hi) the system (U,B,C) is detectable and the right input-output operator % is (V(R+,X),V(R+,Y))-stable; (iv) the system (U,B,C) is stabilizable, detectable and the global input-output operator S is (V(K + ,U),V(K + ,Y))-stable. Proof. This follows using Corollary 3.5 and similar arguments with those from the proof of Theorem 4.9. □ 5. Conclusions Stabilizability and detectability of variational /nonautonomous control systems are two properties which are strongly related with the stable behavior of the initial integral system. These two properties (not even together) cannot assure the uniform exponential stability of the initial system, as Example 3.1 shows. But, in association with a stability of certain input-output operators the stabilizability or /and the detectability of the control system (tz,B,C) imply 454 Robust Control, Theory and Applications the existence of the exponentially stable behavior of the initial system (S n ). Here we have extended the topic from evolution families to variational systems and the obtained results are given in a more general context. As we have shown in Remark 2.6 the spaces involved in the stability properties of the associated input-output operators may be not only V 3 -spaces but also general Orlicz function spaces which is an aspect that creates an interesting link between the modern control theory of dynamical systems and the classical interpolation theory. It worth mentioning that the framework presented in this chapter may be also extended to some slight weak concepts, taking into account the main results concerning the uniform stability concept from Section 3 in (Sasu, 2008) (see Definition 3.3 and Theorem 3.6 in (Sasu, 2008)). More precisely, considering that the system (n,B,C) is weak stabilizable (respectively weak detectable) if there exists a mapping F G C S (®,B(X,U)) (respectively K G C S (®,B(Y,X))) such that the system (S nBF ) (respectively (S nKC )) is uniformly stable, then starting with the result provided by Theorem 3.6 in (Sasu, 2008), the methods from the present chapter may be applied to the study of the uniform stability in terms of weak stabilizability and weak detectability. In authors opinion, the technical trick of the new study will rely on the fact that in this case the families of the associated input-output operators will have to be (L 1 ,L°°)-stable. 6. References Bennett, C. & Sharpley, R. (1988). Interpolation of Operators, Pure Appl. Math. 129, ISBN 0-12-088730-4 Clark, S.; Latushkin, Y.; Montgomery-Smith, S. & Randolph, T. (2000). Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach, SIAM J. Control Optim. Vol. 38, 1757-1793, ISSN 0363-0129 Curtain, R. & Zwart, H. J. (1995). An Introduction to Infinite-Dimensional Linear Control Systems Theory, Springer- Verlag, New- York, ISBN 0-387-94475-3 Megan, M; Sasu, A. L. & Sasu, B. (2002). Stabilizability and controllability of systems associated to linear skew-product semiflows, Rev. Mat. Complutense (Madrid) Vol. 15, 599-618, ISSN 1139-1138 Megan, M.; Sasu, A. L. & Sasu, B. (2005). Theorems of Perron type for uniform exponential stability of linear skew-product semiflows, Dynam. Contin. Discrete Impulsive Systems Vol. 12, 23-43, ISSN 1201-3390 Sasu, B. & Sasu, A. L. (2004). Stability and stabilizability for linear systems of difference equations, J. Differ. Equations Appl. Vol. 10, 1085-1105, ISSN 1023-6198 Sasu, B. (2008). Robust stability and stability radius for variational control systems, Abstract Appl. Analysis Vol. 2008, Article ID 381791, 1-29, ISSN 1085-3375 20 Robust Linear Control of Nonlinear Flat Systems Hebertt Sira-Ramirez 1 , John Cortes-Romero 1,2 and Alberto Luviano-Juarez 1 1 Cinvestav IPN, Av. IPNNo. 2508, Departamento de Ingenieria Electrica, Section de Mecatronica 2 Universidad National de Colombia. Facultad de Ingenieria, Departamento de Ingenieria Electrica y Electronica. Carrera 30 No. 45-03 Bogota, Colombia 1 Mexico 2 Colombia 1. Introduction Asymptotic estimation of external, unstructured, perturbation inputs, with the aim of exactly, or approximately, canceling their influences at the controller stage, has been treated in the existing literature under several headings. The outstanding work of professor CD. Johnson in this respect, under the name of Disturbance Accommodation Control (DAC), dates from the nineteen seventies (see Johnson (1971)). Ever since, the theory and practical aspects of DAC theory have been actively evolving, as evidenced by the survey paper by Johnson Johnson (2008). The theory enjoys an interesting and useful extension to discrete-time systems, as demonstrated in the book chapter Johnson (1982). In a recent article, by Parker and Johnson Parker & Johnson (2009), an application of DAC is made to the problem of decoupling two nonlinear ly coupled linear systems. An early application of disturbance accommodation control in the area of Power Systems is exemplified by the work of Mohadjer and Johnson in Mohadjer & Johnson (1983), where the operation of an interconnected power system is approached from the perspective of load frequency control. A closely related vein to DAC is represented by the sustained efforts of the late Professor Jingqing Han, summarized in the posthumous paper, Han Han (2009), and known as: Active Disturbance Estimation and Rejection (ADER). The numerous and original developments of Prof. Han, with many laboratory and industrial applications, have not been translated into English and his seminal contributions remain written in Chinese (see the references in Han (2009)). Although the main idea of observer-based disturbance estimation, and subsequent cancelation via the control law, is similar to that advocated in DAC, the emphasis in ADER lies, mainly, on nonlinear observer based disturbance estimation, with necessary developments related to: efficient time derivative computation, practical relative degree computation and nonlinear PID control extensions. The work, and inspiration, of Professor Han has found interesting developments and applications in the work of Professor Z. Gao and his colleagues ( see Gao et al. (2001), Gao (2006), also, in the work by Sun and Gao Sun & Gao (2005) and in the article by Sun Sun (2007)). In a recent article, a closely related idea, proposed by Prof. M. Fliess and C. Join in Fliess & Join (2008), is at the core of Intelligent PID ControlQPlDC). The mainstream of the IPIDC developments makes use of the Algebraic Method and it implies to resort to first order, or at most second order, non-phenomenological plant models. The interesting aspect of this method resides in using suitable algebraic manipulations to 456 Robust Control, Theory and Applications locally deprive the system description of the effects of nonlinear uncertain additive terms and, via further special algebraic manipulations, to efficiently identify time-varying control gains as piece-wise constant control input gains (see Fliess et al. (2008)). An entirely algebraic approach for the control of synchronous generator was presented in Fliess and Sira-Ramirez, Sira-Ramirez & Fliess (2004). In this chapter, we advocate, within the context of trajectory tracking control for nonlinear flat systems, the use of approximate, yet accurate, state dependent disturbance estimation via linear Generalized Proportional Integral (GPI) observers. GPI observers are the dual counterpart of GPI controllers, developed by M. Fliess et al. in Fliess et al. (2002). A high gain GPI observer naturally includes a, self-updating, lumped, time-polynomial model of the nonlinear state-dependent perturbation; it estimates it and delivers the time signal to the controller for on-line cancelation while simultaneously estimating the phase variables related to the measured output. The scheme is, however, approximate since only a small as desired reconstruction error is guaranteed at the expense of high, noise-sensitive, gains. The on-line approximate estimation is suitably combined with linear, estimation-based, output feedback control with the appropriate, on-line, disturbance cancelation. The many similarities and the few differences with the DAC and ADER techniques probably lie in 1) the fact that we do not discriminate between exogenous (i.e., external) unstructured perturbation inputs and endogenous (i.e., state-dependent) perturbation inputs in the nonlinear input-output model. These perturbations are all lumped into a simplifying time-varying signal that needs to be linearly estimated. Notice that plant nonlinearities generate time functions that are exogenous to any observer and, hence, algebraic loops are naturally avoided 2) We emphasize the natural possibilities of differentially flat systems in the use of linear disturbance estimation and linear output feedback control with disturbance cancelation (For the concept of flatness see Fliess et al Fliess et al. (1995)) and the book Sira-Ramirez & Agrawal (2004). This chapter is organized as follows: Section 2 presents an introduction to linear control of nonlinear differentially flat systems via (high-gain) GPI observers and suitable linear controllers feeding back the phase variables related to the output function. The single input-single output synchronous generator model in the form a swing equation, is described in Section 3. Here, we formulate the reference trajectory tracking problem under a number of information restrictions about the system. The linear observer-linear controller output feedback control scheme is designed for lowering the deviation angle of the generator. We carry out a robustness test regarding the response to a three phase short circuit. We also carry an evaluation of the performance of the control scheme under significant variations of the two control gain parameters required for an exact cancelation of the gain. Section 4 is devoted to present an experimental illustrative example concerning the non-holonomic car which is also a multivariable nonlinear system with input gain matrix depending on the estimated phase variables associated with the flat outputs. 2. Linear GPI observer-based control of nonlinear systems Consider the following perturbed nonlinear single-input single input-output, smooth, nonlinear system, yW=xp(t,y,y y^) +cp{t,y)u + £(f) (1) The unperturbed system, (£(£) = 0) is evidently flat, as all variables in the system are expressible as differential functions of the flat output y. We assume that the exogenous perturbation £(£) is uniformly absolutely bounded, i.e., it an Loo scalar function. Similarly, we assume that for all bounded solutions, y(t), of (1), Robust Linear Control of Nonlinear Flat Systems 457 obtained by means of suitable control input u, the additive, endogenous, perturbation input, xp(t,y(t),y(t),...,y( n ~ 1 \t)), viewed as a time signal is uniformly absolutely bounded. We also assume that the nonlinear gain function (p(t, y (t ) ) is Loo and uniformly bounded away from zero, i.e., there exists a strictly positive constant ]i such that inf|<Hf,J/(f))|> \i (2) for all smooth, bounded solutions, y(t), of (1) obtained with a suitable control input u. Although the results below can be extended when the input gain function (p depends on the time derivatives of y, we let, motivated by the synchronous generator case study to be presented, (p to be an explicit function of time and of the measured flat output y. This is equivalent to saying the (p(t,y(t)) is perfectly known. We have the following formulation of the problem: Given a desired flat output reference trajectory, y*(t), devise a linear output feedback controller for system (1) so that regardless of the endogenous perturbation signal ip(t,y(t),y(t), ...,y( n ~ l \t)) and of the exogenous perturbation input £(£), the flat output y tracks the desired reference signal y* (t) even if in an approximate fashion. This approximate character specifically means that the tracking error, e (0 = V ~ 3/*(0/ m & its first, n, time derivatives, globally asymptotically exponentially converge towards a small as desired neighborhood of the origin in the reference trajectory tracking error phase space. The solution to the problem is achieved in an entirely linear fashion if one conceptually considers the nonlinear model (1) as the following linear perturbed system yW=v + S(t) (3) where v = <p{t,y)u, and £(*) = tp(t,y(t),y(t), ...y^" 1 ^)) + £(*)■ Consider the following preliminary result: Proposition 1. The unknown perturbation vector of time signals, %(t), in the simplified tracking error dynamics (3), is observable in the sense ofDiop and Fliess (see Diop & Fliess (1991))). Proof The proof of this fact is immediate after writing (3) as m = y (n) -v = y {n) -<p{t,y)u (4) i.e., £(£) can be written in terms of the output vector y, a finite number of its time derivatives and the control input u. Hence, £(£) is observable. Remark 2. This means, in particular, that if g(t) is bestowed with an exact linear model; an exact asymptotic estimation of %{i) is possible via a linear observer. If, on the other hand, the linear model is only approximately locally valid, then the estimation obtained via a linear observer is asymptotically convergent towards an equally approximately locally valid estimate. We assume that the perturbation input g(t) may be locally modeled as a p — 1-th degree time polynomial z\ plus a residual term, r(t), i.e., £(t) = z 1 + r(t) = a + a x t + • • • + flp-i^" 1 + r(t), for all t (5) The time polynomial model, Z\, (also called: a Taylor polynomial) is invariant with respect to time shifts and it defines a family of p — 1 degree Taylor polynomials with arbitrary real 458 Robust Control, Theory and Applications coefficients. We incorporate Z\ as an internal model of the additive perturbation input (see Johnson (1971)). The perturbation model Z\ will acquire a self updating character when incorporated as part of a linear asymptotic observer whose estimation error is forced to converge to a small vicinity of zero. As a consequence of this, we may safely assume that the self-updating residual function, r(t ), and its time derivatives, say r^\t), are uniformly absolutely bounded. To precisely state this, let us denote by yy an estimate of y^' -1 ) for ;' = 1, ..., n. We have the following general result: Theorem 3. The GPI observer-based dynamical feedback controller: n-\ <p(t,y) [y*W] (n) -Eh-[y;-(y*W) (;) ])-ew ;=0 1(0 = *1 (6) Vi =y2 + A p+n _i(y-yi) y 2 = y 3 + A p+n _ 2 (y-yi) y n = p + z 1 +A p (y-y 1 ) Z! = Z2 + A p _i(y-yi) Zp-i = z p + M(y-yi) Zp = k (y-y 1 ) (7) asymptotically exponentially drives the tracking error phase variables, e\ ' = y^> — [y*(0] / k = 0,1,.. ,n — 1 to an arbitrary small neighborhood of the origin, of the tracking error phase space, which can be made as small as desired from the appropriate choice of the controller gain parameters {kq, ...,K n _i}. Moreover, the estimation errors: e\ l > = y^ l > — y z -, i = 0, ...,n — 1 and the perturbation estimation error: z m — £ m_1 (£), m = l,...,p asymptotically exponentially converge towards a small as desired neighborhood of the origin of the reconstruction error space which can be made as small as desired from the appropriate choice of the controller gain parameters {Aq, ..., A p+H _i}. Proof The proof is based on the fact that the estimation error e satisfies the perturbed linear differential equation ^+") + A p+n _ig^ +n - 1 ) + • • • + \ e = r {v) (t) (8) Since r^\t) is assumed to be uniformly absolutely bounded then there exists coefficients A^ such that e converges to a small vicinity of zero, provided the roots of the associated characteristic polynomial in the complex variable s: sP +n + Ap+n-is^"- 1 + • • • + Ais + A (9) Robust Linear Control of Nonlinear Flat Systems 459 are all located deep into the left half of the complex plane. The further away from the imaginary axis, of the complex plane, are these roots located, the smaller the neighborhood of the origin, in the estimation error phase space, where the estimation error e will remain ultimately bounded (see Kailath Kailath (1979)). Clearly, if e and its time derivatives converge to a neighborhood of the origin, then Zj — £u\ j = 1,2,..., also converge towards a small vicinity of zero. The tracking error ey = y — y*(t) evolves according to the following linear perturbed dynamics 4 B) + *»-i4" _1) + • • • + K ° e v = fto - ft') < 10 ) Choosing the controller coefficients {kq, • • • ,K n _i}, so that the associated characteristic polynomial S n +!C n _iS n - 1 + -..+K0 (11) exhibits its roots sufficiently far from the imaginary axis in the left half portion of the complex plane, the tracking error, and its various time derivatives, are guaranteed to converge asymptotically exponentially towards a vicinity of the tracking error phase space. Note that, according to the observer expected performance, the right hand side of (10) is represented by a uniformly absolutely bounded signal already evolving on a small vicinity of the origin. For this reason the roots of (11) may be located closer to the imaginary axis than those of (9). A rather detailed proof of this theorem may be found in the article by Luviano et al. Luviano-Juarez et al. (2010) Remark 4. The proposed GPI observer (7) is a high gain observer which is prone to exhibiting the "peaking" phenomena at the initial time. We use a suitable "clutch" to smooth out these transient peaking responses in all observer variables that need to be used by the controller. This is accomplished by means of a factor function smoothly interpolating between an initial value of zero and a final value of unity. We denote this clutching function as sAt) <G [0, 1] and define it in the following (non-unique) way / n _ / 1 for t > e S / (t) -\sin4(g)forf<e (12) where a is a su itably large positive even integer. 2.1 Generalized proportional integral observer with integral injection Let £(£) be a measured signal with an uniformly absolutely bounded iterated integral of order m. The function g(t) is a measured signal, whose first few time derivatives are required for some purpose. Definition 5. We say that a signal p\(t) converges to a neighborhood of g(t) whenever the error signal, £(t) — pi (t), is ultimately uniformly absolutely bounded inside a small vicinity of the origin. The following proposition aims at the design of a GPI observer based estimation of time derivatives of a signal, £(£), where £(t) is possibly corrupted by a zero mean stochastic process whose statistics are unknown. In order to smooth out the noise effects on the on-line computation of the time derivative, we carry out a double iterated integration of the measured signal, £(£), thus assuming the second integral of £(t) is uniformly absolutely bounded (i.e., m = 2). 460 Robust Control, Theory and Applications Proposition 6. Consider the following perturbed second order integration system, where the input signal, %(t),isa measured (zero-mean) noise corrupted signal satisfying the above assumptions: Vo = 3/1/ Vl = S(0 (13) Consider the following integral injection GPI observer for (13) including an internal time polynomial model of degree r for the signal £(£) and expressed as pi, h = yi + A r+1 (y -y ) fa =pi + A r (yo-yo) pi =p2 + A r _i(y -y ) (14) pr = A (y -yo) (15) Then, the observer variables, p\, p2, P3,..., respectively, asymptotically converge towards a small as desired neighborhood of the disturbance input, £(£), and of its time derivatives: £(£), £(*)/••• provided the observer gains, { Aq, ..., A r+ 2}/ are chosen so that the roots of the polynomial in the complex variable s. P(s) = s r+2 + A r+1 s r+1 + • • • + Ais + A (16) are located deep into the left half of the complex plane. The further the distance of such roots from the imaginary axis of the complex plane, the smaller the neighborhood of the origin bounding the reconstruction errors. Proof. Define the twice iterated integral injection error as, e = yo — t/rj. The injection error dynamics is found to be described by the perturbed linear differential equation £ ( r+2 ) + A r+ iz {r+1) + • • ■ + Aie + A £ = £« (t) (17) By choosing the observer parameters, Aq, Ai, • • • , A r+ i, so that the polynomial (16) is Hurwitz, with roots located deep into the left half of the complex plane, then, according to well known results of solutions of perturbed high gain linear differential equations, the injection error e and its time derivatives are ultimately uniformly bounded by a small vicinity of the origin of the reconstruction error phase space whose radius of containment fundamentally depends on the smallest real part of all the eigenvalues of the dominantly linear closed loop dynamics (see Luviano et al. Luviano-Juarez et al. (2010) and also Fliess and Rudolph Fliess & Rudolph (1997)). □ 3. Controlling the single synchronous generator model In this section, we advocate, within the context of the angular deviation trajectory control for a single synchronous generator model, the use of approximate, yet accurate, state dependent disturbance estimation via linear Generalized Proportional Integral (GPI) observers. GPI observers are the dual counterpart of GPI controllers, developed by M. Fliess et al. in Fliess et al. (2002). A high gain GPI observer naturally includes a, self-updating, lumped, time-polynomial model of the nonlinear state-dependent perturbation; it estimates it and delivers the time signal to the controller for on-line cancelation while simultaneously estimating the phase variables related to the measured output. The scheme is, however, approximate since only a small as desired reconstruction error is guaranteed at the expense Robust Linear Control of Nonlinear Flat Systems 461 of high, noise-sensitive, gains. The on-line approximate estimation is suitably combined with linear, estimation-based, output feedback control with the appropriate, on-line, disturbance cancelation. The many similarities and the few differences with the DAC and ADER techniques probably lie in 1) the fact that we do not discriminate between exogenous (i.e., external) unstructured perturbation inputs and endogenous (i.e., state-dependent) perturbation inputs in the nonlinear input-output model. These perturbations are all lumped into a simplifying time-varying signal that needs to be linearly estimated. Notice that plant nonlinearities generate time functions that are exogenous to any observer and, hence, algebraic loops are naturally avoided 2) We emphasize the natural possibilities of differentially flat systems in the use of linear disturbance estimation and linear output feedback control with disturbance cancelation (For the concept of flatness see Fliess et al. Fliess et al. (1995)) and the book Sira-Ramirez & Agrawal (2004). 3.1 The single synchronous generator model Consider the swing equation of a synchronous generator, connected to an infinite bus, with a series capacitor connected with the help of a thyristor bridge (See Hingorani Hingorani & Gyugyi (2000)), xi = x 2 x 2 = P m ~ b x x 2 - b 2 x 3 sin(*i) *3 = h(-X3+4(*) + u + tt t )) ( 18 ) X\ is the load angle, considered to be the measured output. The variable, x 2 , is the deviation from nominal, synchronous, speed at the shaft, while X3 stands for the admittance of the system. The control input, u, is usually interpreted as a quantity related to the fire angle of the switch. £(t) is an unknown, external, perturbation input. The static equilibrium point of the system, which may be parameterized in terms of the equilibrium position for the angular deviation, X\, is given by, *i=si,x 2 = o,S3 = 3(0 = a^Efe < 19) We assume that the system parameters, b 2/ and, b$, are known. The constant quantities P m , b\ and the time varying quantity, x^(t), are assumed to be completely unknown. 3.2 Problem formulation It is desired to have the load angular deviation, y = x\, track a given reference trajectory, y*(t) = x\ (t), which remains bounded away from zero, independently of the unknown system parameters and in spite of possible external system disturbances (such as short circuits in the three phase line, setting, momentarily, the mechanical power, P m , to zero), and other unknown, or un-modeled, perturbation inputs comprised in £(£). 3.3 Main results The unperturbed system in (18) is flat, with flat output given by the load angle deviation y = x\ . Indeed, all system variables are differentially parameterizable in terms of the load 462 Robust Control, Theory and Applications angle and its time derivatives. We have: *l =y *2 = y P m -hy-y *3 fr 2 sin(y) hy + yW P m -hy-y b 3 b 2 sin(y) b 3 b 2 sin 2 (y) ycos(y) + P -^y-y _ fr 2 sm(y) JW The perturbed input-output dynamics, devoid of any zero dynamics, is readily obtained with the help the control input differential parametrization (20). One obtains the following simplified, perturbed, system dynamics, including £(*), as: (21) y(V = -[b 3 b 2 sm(y)}u + Z(t) £(£) is given by m = *"ft-w-»('-d^ -&3&2sin(i/)(x|(f)+^(f)) (22) We consider %(t) as an unknown but uniformly absolutely bounded disturbance input that needs to be on-line estimated by means of an observer and, subsequently, canceled from the simplified system dynamics via feedback in order to regulate the load angle variable y towards the desired reference trajectory y*(t). It is assumed that the gain parameters b 2 and b 3 are known. The problem is then reduced to the trajectory tracking problem defined on the perturbed third order, predominantly, linear system (21) with measurable state dependent input gain and unknown, but uniformly bounded, disturbance input. We propose the following estimated state feedback controller with a smoothed (i.e., "clutched" ) disturbance cancelation term, z\ s (i) = Sf(t)z\(t), and smoothed estimated phase variables yjs — s /(03/;(0/ ] = 1/ 2, 3 with Sf(t) as in equation (12) with a suitable e value. [(y*W) (3) -^(y3s-y*W) b 3 b 2 sin(y) -h(yis - fit)) - k (y - y*(0) - z ls ] The corresponding variables, y 3 , y 2 and Z\, are generated by the following linear GPI observer: y\ = y2 + A 5 (y-yi) y 2 = y3 + A 4 (y-yi) 1/3 = -(&3&2sin(y))tt + Zi + A 3 (i/-i/i) z 1 = z 2 + A 2 (y-yi) z 2 = z 3 + Ai(y-yi) z 3 = Ao(y-yi) (23) Robust Linear Control of Nonlinear Flat Systems 463 where y\ is the redundant estimate of the output y, y 2 is the shaft velocity estimate and 1/3 is the shaft acceleration estimate. The variable z\ estimates the perturbation input £(£) by means of a local, self updating, polynomial model of third order, taken as an internal model of the state dependent additive perturbation affecting the input-output dynamics (21). The clutched observer variables Z\ s , y 2s and y 3s are defined by «.=»/(^-/w={f 8(S) s;^ (24) with S being either z\ s , y 2s or V?>s The reconstruction error system is obtained by subtracting the observer model from the perturbed simplified linear system model. We have, letting e = e\ = y — \j\, e 2 = y — y 2/ etc. h = e 2 - \$e\ e 2 = e 3 - \ 4: e 1 h = ?(0 -zi- ^1 z\ = z2 + A 2 (y-yi) Z2 = Z3 + Ai(y-yi) Z3 : = Ao(y-yi) (25) The reconstruction dynamics error, e = e 1 = y - ] + A 5 e^ + A. - yi, is seen to satisfy i^ + '-' + A^+Aoe the following linear, perturbed, (26) Choosing the gains {A 5 ,- •-,A } so that the roots of the characteristic polynomial, Po(s) = s 6 + A 5 s 5 + A 4 s 4 + --- + Aif i + A , (27) are located deep into the left half of the complex plane, it follows from the bounded input, bounded output stability theory that the trajectories of the reconstruction error e and those of its time derivatives e^\ j = 1, 2, ... are uniformly ultimately bounded by a disk, centered at the origin in the reconstruction error phase space, whose radius can be made arbitrarily small as the roots of p (s) are pushed further to the left of the complex plane. The closed loop tracking error dynamics satisfies 4 3) + ^4 2) + My + K e y = £(*) - z ls (28) The difference, £(t) — Z\ s , being arbitrarily small after some time, produces a reference trajectory tracking error, e y = y — y*(t), that also asymptotically exponentially converges towards a small vicinity of the origin of the tracking error phase space. The characteristic polynomial of the predominant linear component of the closed loop system may be set to have poles placed in the left half of the complex plane at moderate locations p c (S) = S 3 + K 2 S 2 + KiS + K (29) 464 Robust Control, Theory and Applications 3.4 Simulation results 3.4.1 A desired rest-to-rest maneuver It is desired to smoothly lower the load angle, \)\ = X\, from an equilibrium value of y = 1 [rad] towards a smaller value, say, y = 0.6 [rad] in a reasonable amount of time, say, T = 5 [s], starting at t = 5 [s] of an equilibrium operation characterized by (see Bazanella et ah Bazanella et al. (1999) and Pai Pai (1989)) x 1 = 1, x 2 = 0, x 3 = 0.8912 We used the following parameter values for the system h = 1, b 2 = 21.3360, b 3 = 20 We set the external perturbation input, £,(t ), as the time signal, £(0 = 0.005^ sin2 ^) cos ^))cos(0.30 The observer parameters were set in accordance with the following desired characteristic polynomial p (s) for the, predominantly, linear reconstruction error dynamics. We set p (s) = (s 2 + 2l, cv no s + o; 2 ) 3 , with £ = 1, cv no = 20 The controller gains K2,K\,kq were set so that the following closed loop characteristic polynomial, p c (s), was enforced on the tracking error dynamics, p c (s) = (s 2 + 2l, c cv nc s + cvl c )(s + p c ) with p c = 3, COnc = 3/ £c = 1 The trajectory for the load angle, y*(t), was set to be y* 0) = *l,initial + (p(t, h, h)) (^l,final ~ ^initial) with p{t,t\,t2) being a smooth Bezier polynomial achieving a smooth rest-to-rest trajectory for the nominal load angle y* (t) from the initial equilibrium value y* (t\) = x linitia i = 1 [rad] towards the final desired equilibrium value y*(^) — ^i,final — ^.6 [rad]. We set t\ = 5.0 [s], t 2 = 10.0 [s]; e = 3.0 The interpolating polynomial p(t, t\, t 2 ), is of the form: p(t) = T 8 fi — f2T + f3T 2 — r 4 T 3 + f5T 4 -r 6 r 5 + r 7 T 6 - r 8 r 7 + r 9 r^ with, The choice, h-h r x = 12870, r 2 = 91520, r 3 = 288288 r 4 = 524160, r 5 = 600600, r 6 = 443520 r 7 = 205920, r 8 = 54912, r 9 = 6435 Robust Linear Control of Nonlinear Flat Systems 465 1.5 1 0.5 -0.5 Xl{t) V x 3 (t) x 2 (t) V(t),y*(t) 1 0.8 0.6 10 15 10 15 0.02 0.01 -0.01 10 15 10 15 Fig. 1. Performance of GPI observer based linear controller for load angle rest-to-rest trajectory tracking in a perturbed synchronous generator. renders a time polynomial which is guaranteed to have enough derivatives being zero, both, at the beginning and at the end of the desired rest to rest maneuver. Figure 1 depicts the closed loop performance of the proposed GPI observer based linear output feedback controller for the forced evolution of the synchronous generator load angle trajectory following a desired rest to rest maneuver. 3.4.2 Robustness with respect to controller gain mismatches We simulated the behavior of the closed loop system when the gain parameters product, b^b^, is not precisely known and the controller is implemented with an estimated (guessed) value of this product, denoted by ^3/ an d set to be ^3 = ^2^3- We determined that k is a positive factor ranging in the interval [0.95, oo] . However, if we allow independent estimates of the parameters in the form b-i = k^ 2 ^2 an d b$ = K^b^, we found that a larger robustness interval of mismatches is allowed by satisfying the empirical relation k^k^ > 0.95. The assessment 466 Robust Control, Theory and Applications 1.2 1 1.1 Xi(t) 0.8 1 0.6 0.4 0.9 0.8 0.2 10 Fig. 2. Performance of GPI observer based controller under a sudden loss of power at t=2 [sec] during 0.2 [sec]. was made in terms of the proposed rest to rest maneuver and possible simulations look about the same. 3.4.3 Robustness with respect to sudden power failures We simulated an un-modeled sudden three phase short circuit occurring at time t = 2 [s]. The power failure lasts for t = 0.2 [s]. Figure 3 depicts the performance of the GPI observer based controller in the rapid transient occurring during the recovery of the prevailing equilibrium conditions. 4. Controlling the non-holonomic car Controlling non-holonomic mobile robots has been an active topic of research during the past three decades due to the wide variety of applications. Several methods have been proposed, and applied, to solve the regulation and trajectory tracking tasks in mobile robots. These methods range from sliding mode techniques Aguilar et al. (1997), Wang et al. (2009), Robust Linear Control of Nonlinear Flat Systems 467 Yang & Kim (1999), backstepping Hou et al. (2009), neural networks approaches (see Peng et al. (2007) and references therein), linearization techniques Kim & Oh (1999), and classical control approaches (see Sugisaka & Hazry (2007)) among many other possibilities. A classical contribution to this area is given in the work of Canudas de Wit Wit & Sordalen (1992). An excellent book, dealing with some appropriate control techniques for this class of systems, is that of Dixon et al. Dixon et al. (2001). A useful approach to control non-holonomic mechanical systems is based on linear time-varying control schemes (see Pomet (1992); Tian & Cao (2007)). In the pioneering work of Samson Samson (1991), smooth feedback controls (depending on an exogeneous time variable) are proposed to stabilize a wheeled cart. It has been shown that some mobile robotic systems are differentially flat when slippage is not allowed in the model ( see Leroquais & d' Andrea Novel (1999)). The differential flatness property allows a complete parametrization of all system variables in terms of the flat outputs an a and a finite number of their time derivatives. Flat outputs constitute a limited set of special, differentially independent, output variables. The reader is referred to the work of Fliess et al. Fliess et al. (1995) for the original introduction of the idea in the control systems literature. From the flatness of the non-holonomic car system, it is possible to reduce the control task to that of a linearizable, extended, multivariable input-output system. The linearization of the flat output dynamics requires the cancelation of the nonlinear input gain matrix, which depends only on the cartesian velocities of the car. To obtain this set of noisy unmeasured state variables, we propose linear Generalized Proportional Integral (GPI) observers consisting of linear, high gain Luenberger-like observers Luenberger (1971) exhibiting an internal polynomial model for the measured signal. These GPI observers, introduced in Sira-Ramirez & Feliu-Battle (2010), can provide accurate, filtered, time derivatives of the injected output signals via an appropriate iterated integral estimation error injection (see also Cortes-Romero et al. (2009)). Since high-gain observers are known to be sensitive to noisy measurements, the iterated integral injection error achieves a desirable low pass filtering effect. The idealized model of a single axis two wheeled vehicle is depicted in figure 3. The axis is of length L and each wheel of radius R is powered by a direct current motor yielding variable angular speeds co\, CV2 respectively. The position variables are (x\, X2) and 6 is the orientation angle of the robot. The linear velocities of the points of contact of the wheels respect to the ground are given by v\ = coiR and vi = o^- m this case, the only measurable variables are X\, X2- This system is subject to non-holonomic restrictions. The kinematic model of the system is stated as follows (30) where: jR/2 R/2 -R/L R/L cv 1 0J2 The control objective is stated as follows: given a desired trajectory (x\{t) , x^it)) , devise feedback control laws, U\, ui, such that the flat output coordinates, [x\,X2), perform an asymptotic tracking while rejecting the un-modeled additive disturbances. 468 Robust Control, Theory and Applications x 2 * v\ R //v/ X! Fig. 3. The one axis car 4.1 Controller design System (30) is differentially flat, with flat outputs given by the pair of coordinates: (x\,X2), which describes the position of the rear axis middle point. Indeed the rest of the system variables, including the inputs are differentially parameterized as follows: 6 = arctan x\ + x\, ui ■ X 2 X\ — XiX\ X-l I Jir\ Note that the relation between the inputs and the flat outputs highest derivatives is not invertible due to an ill defined relative degree. To overcome this obstacle to feedback linearization, we introduce, as an extended auxiliary control input, the time derivative of u\. We have: u\ X\X\ + x 2 x 2 s + x$ This control input extension yields now an invertible control input-to-flat outputs highest derivatives relat